pydrake.trajectories

class pydrake.trajectories.BezierCurve

Bases: pydrake.trajectories.Trajectory

A Bézier curve is defined by a set of control points p₀ through pₙ, where n is called the order of the curve (n = 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the endpoints of the curve; however, the intermediate control points (if any) generally do not lie on the curve, but the curve is guaranteed to stay within the convex hull of the control points.

See also BsplineTrajectory. A B-spline can be thought of as a composition of overlapping Bézier curves (where each evaluation only depends on a local subset of the control points). In contrast, evaluating a Bézier curve will use all of the control points.

Note

This class is templated; see BezierCurve_ for the list of instantiations.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.BezierCurve) -> None

Default initializer. Constructs an empty Bézier curve over the interval t ∈ [0, 1].

__init__(self: pydrake.trajectories.BezierCurve, start_time: float, end_time: float, control_points: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) -> None

Constructs a Bézier curve over the interval t ∈ [start_time, end_time`] with control points defined in the columns of `control_points.

Precondition:: end_time >= start_time.

AsLinearInControlPoints(self: pydrake.trajectories.BezierCurve, derivative_order: int = 1) → scipy.sparse.csc_matrix[numpy.float64]

Supports writing optimizations using the control points as decision variables. This method returns the matrix, M, defining the control points of the order derivative in the form:

Click to expand C++ code...

derivative.control_points() = this.control_points() * M

For instance, since we have

Click to expand C++ code...

derivative.control_points().col(k) = this.control_points() * M.col(k),

constraining the kth control point of the `n`th derivative to be in [ub, lb], could be done with:

Click to expand C++ code...

auto M = curve.AsLinearInControlPoints(n);
for (int i=0; i<curve.rows(); ++i) {
auto c = std::make_shared<solvers::LinearConstraint>(
M.col(k).transpose(), Vector1d(lb(i)), Vector1d(ub(i)));
prog.AddConstraint(c, curve.row(i).transpose());
}

Iterating over the rows of the control points is the natural sparsity pattern here (since M is the same for all rows). For instance, we also have

Click to expand C++ code...

derivative.control_points().row(k).T = M.T * this.control_points().row(k).T,

or

Click to expand C++ code...

vec(derivative.control_points().T) = blockMT * vec(this.control_points().T),
blockMT = [ M.T,   0, .... 0 ]
[   0, M.T, 0, ... ]
[      ...         ]
[  ...    , 0, M.T ].

Precondition:: derivative_order >= 0.

BernsteinBasis(self: pydrake.trajectories.BezierCurve, i: int, time: float, order: Optional[int] = None) → float: Returns the value of the ith basis function of order (1 for linear, 2 for quadratic, etc) evaluated at time. The default value for the optional argument order is the order() of this.

control_points(self: pydrake.trajectories.BezierCurve) → numpy.ndarray[numpy.float64[m, n]]: Returns a const reference to the control points which define the curve.

ElevateOrder(self: pydrake.trajectories.BezierCurve) → None: Increases the order of the curve by 1. A Bézier curve of order n can be converted into a Bézier curve of order n + 1 with the same shape. The control points of this are modified to obtain the equivalent curve.

GetExpression(self: pydrake.trajectories.BezierCurve, time: pydrake.symbolic.Variable = Variable('t', Continuous)) → numpy.ndarray[object[m, 1]]: Extracts the expanded underlying polynomial expression of this curve in terms of variable time.

order(self: pydrake.trajectories.BezierCurve) → int: Returns the order of the curve (1 for linear, 2 for quadratic, etc.).

template pydrake.trajectories.BezierCurve_: Instantiations: BezierCurve_[float], BezierCurve_[AutoDiffXd], BezierCurve_[Expression]

class pydrake.trajectories.BezierCurve_[AutoDiffXd]

Bases: pydrake.trajectories.Trajectory_[AutoDiffXd]

A Bézier curve is defined by a set of control points p₀ through pₙ, where n is called the order of the curve (n = 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the endpoints of the curve; however, the intermediate control points (if any) generally do not lie on the curve, but the curve is guaranteed to stay within the convex hull of the control points.

See also BsplineTrajectory. A B-spline can be thought of as a composition of overlapping Bézier curves (where each evaluation only depends on a local subset of the control points). In contrast, evaluating a Bézier curve will use all of the control points.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.BezierCurve_[AutoDiffXd]) -> None

Default initializer. Constructs an empty Bézier curve over the interval t ∈ [0, 1].

__init__(self: pydrake.trajectories.BezierCurve_[AutoDiffXd], start_time: float, end_time: float, control_points: numpy.ndarray[object[m, n], flags.f_contiguous]) -> None

Constructs a Bézier curve over the interval t ∈ [start_time, end_time`] with control points defined in the columns of `control_points.

Precondition:: end_time >= start_time.

AsLinearInControlPoints(self: pydrake.trajectories.BezierCurve_[AutoDiffXd], derivative_order: int = 1) → scipy.sparse.csc_matrix[numpy.float64]

Supports writing optimizations using the control points as decision variables. This method returns the matrix, M, defining the control points of the order derivative in the form:

Click to expand C++ code...

derivative.control_points() = this.control_points() * M

For instance, since we have

Click to expand C++ code...

derivative.control_points().col(k) = this.control_points() * M.col(k),

constraining the kth control point of the `n`th derivative to be in [ub, lb], could be done with:

Click to expand C++ code...

auto M = curve.AsLinearInControlPoints(n);
for (int i=0; i<curve.rows(); ++i) {
auto c = std::make_shared<solvers::LinearConstraint>(
M.col(k).transpose(), Vector1d(lb(i)), Vector1d(ub(i)));
prog.AddConstraint(c, curve.row(i).transpose());
}

Iterating over the rows of the control points is the natural sparsity pattern here (since M is the same for all rows). For instance, we also have

Click to expand C++ code...

derivative.control_points().row(k).T = M.T * this.control_points().row(k).T,

or

Click to expand C++ code...

vec(derivative.control_points().T) = blockMT * vec(this.control_points().T),
blockMT = [ M.T,   0, .... 0 ]
[   0, M.T, 0, ... ]
[      ...         ]
[  ...    , 0, M.T ].

Precondition:: derivative_order >= 0.

BernsteinBasis(self: pydrake.trajectories.BezierCurve_[AutoDiffXd], i: int, time: pydrake.autodiffutils.AutoDiffXd, order: Optional[int] = None) → pydrake.autodiffutils.AutoDiffXd: Returns the value of the ith basis function of order (1 for linear, 2 for quadratic, etc) evaluated at time. The default value for the optional argument order is the order() of this.

control_points(self: pydrake.trajectories.BezierCurve_[AutoDiffXd]) → numpy.ndarray[object[m, n]]: Returns a const reference to the control points which define the curve.

ElevateOrder(self: pydrake.trajectories.BezierCurve_[AutoDiffXd]) → None: Increases the order of the curve by 1. A Bézier curve of order n can be converted into a Bézier curve of order n + 1 with the same shape. The control points of this are modified to obtain the equivalent curve.

GetExpression(self: pydrake.trajectories.BezierCurve_[AutoDiffXd], time: pydrake.symbolic.Variable = Variable('t', Continuous)) → numpy.ndarray[object[m, 1]]: Extracts the expanded underlying polynomial expression of this curve in terms of variable time.

order(self: pydrake.trajectories.BezierCurve_[AutoDiffXd]) → int: Returns the order of the curve (1 for linear, 2 for quadratic, etc.).

class pydrake.trajectories.BezierCurve_[Expression]

Bases: pydrake.trajectories.Trajectory_[Expression]

A Bézier curve is defined by a set of control points p₀ through pₙ, where n is called the order of the curve (n = 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the endpoints of the curve; however, the intermediate control points (if any) generally do not lie on the curve, but the curve is guaranteed to stay within the convex hull of the control points.

See also BsplineTrajectory. A B-spline can be thought of as a composition of overlapping Bézier curves (where each evaluation only depends on a local subset of the control points). In contrast, evaluating a Bézier curve will use all of the control points.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.BezierCurve_[Expression]) -> None

Default initializer. Constructs an empty Bézier curve over the interval t ∈ [0, 1].

__init__(self: pydrake.trajectories.BezierCurve_[Expression], start_time: float, end_time: float, control_points: numpy.ndarray[object[m, n], flags.f_contiguous]) -> None

Constructs a Bézier curve over the interval t ∈ [start_time, end_time`] with control points defined in the columns of `control_points.

Precondition:: end_time >= start_time.

AsLinearInControlPoints(self: pydrake.trajectories.BezierCurve_[Expression], derivative_order: int = 1) → scipy.sparse.csc_matrix[numpy.float64]

Supports writing optimizations using the control points as decision variables. This method returns the matrix, M, defining the control points of the order derivative in the form:

Click to expand C++ code...

derivative.control_points() = this.control_points() * M

For instance, since we have

Click to expand C++ code...

derivative.control_points().col(k) = this.control_points() * M.col(k),

constraining the kth control point of the `n`th derivative to be in [ub, lb], could be done with:

Click to expand C++ code...

auto M = curve.AsLinearInControlPoints(n);
for (int i=0; i<curve.rows(); ++i) {
auto c = std::make_shared<solvers::LinearConstraint>(
M.col(k).transpose(), Vector1d(lb(i)), Vector1d(ub(i)));
prog.AddConstraint(c, curve.row(i).transpose());
}

Iterating over the rows of the control points is the natural sparsity pattern here (since M is the same for all rows). For instance, we also have

Click to expand C++ code...

derivative.control_points().row(k).T = M.T * this.control_points().row(k).T,

or

Click to expand C++ code...

vec(derivative.control_points().T) = blockMT * vec(this.control_points().T),
blockMT = [ M.T,   0, .... 0 ]
[   0, M.T, 0, ... ]
[      ...         ]
[  ...    , 0, M.T ].

Precondition:: derivative_order >= 0.

BernsteinBasis(self: pydrake.trajectories.BezierCurve_[Expression], i: int, time: pydrake.symbolic.Expression, order: Optional[int] = None) → pydrake.symbolic.Expression: Returns the value of the ith basis function of order (1 for linear, 2 for quadratic, etc) evaluated at time. The default value for the optional argument order is the order() of this.

control_points(self: pydrake.trajectories.BezierCurve_[Expression]) → numpy.ndarray[object[m, n]]: Returns a const reference to the control points which define the curve.

ElevateOrder(self: pydrake.trajectories.BezierCurve_[Expression]) → None: Increases the order of the curve by 1. A Bézier curve of order n can be converted into a Bézier curve of order n + 1 with the same shape. The control points of this are modified to obtain the equivalent curve.

GetExpression(self: pydrake.trajectories.BezierCurve_[Expression], time: pydrake.symbolic.Variable = Variable('t', Continuous)) → numpy.ndarray[object[m, 1]]: Extracts the expanded underlying polynomial expression of this curve in terms of variable time.

order(self: pydrake.trajectories.BezierCurve_[Expression]) → int: Returns the order of the curve (1 for linear, 2 for quadratic, etc.).

class pydrake.trajectories.BsplineTrajectory

Bases: pydrake.trajectories.Trajectory

Represents a B-spline curve using a given basis with ordered control_points such that each control point is a matrix in ℝʳᵒʷˢ ˣ ᶜᵒˡˢ.

See also

math::BsplineBasis

Note

This class is templated; see BsplineTrajectory_ for the list of instantiations.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.BsplineTrajectory) -> None
__init__(self: pydrake.trajectories.BsplineTrajectory, basis: pydrake.math.BsplineBasis, control_points: list[list[float]]) -> None

Constructs a B-spline trajectory with the given basis and control_points.

Precondition:: control_points.size() == basis.num_basis_functions()

__init__(self: pydrake.trajectories.BsplineTrajectory, basis: pydrake.math.BsplineBasis, control_points: list[numpy.ndarray[numpy.float64[m, n]]]) -> None

Constructs a B-spline trajectory with the given basis and control_points.

Precondition:: control_points.size() == basis.num_basis_functions()

basis(self: pydrake.trajectories.BsplineTrajectory) → pydrake.math.BsplineBasis: Returns the basis of this curve.

Clone(self: pydrake.trajectories.BsplineTrajectory) → pydrake.trajectories.Trajectory

control_points(self: pydrake.trajectories.BsplineTrajectory) → list[numpy.ndarray[numpy.float64[m, n]]]: Returns the control points of this curve.

CopyBlock(self: pydrake.trajectories.BsplineTrajectory, start_row: int, start_col: int, block_rows: int, block_cols: int) → pydrake.trajectories.BsplineTrajectory: Returns a new BsplineTrajectory that uses the same basis as this, and whose control points are the result of calling point.block(start_row, start_col, block_rows, block_cols) on each point in this->control_points().

CopyHead(self: pydrake.trajectories.BsplineTrajectory, n: int) → pydrake.trajectories.BsplineTrajectory

Returns a new BsplineTrajectory that uses the same basis as this, and whose control points are the result of calling point.head(n) on each point in this->control_points().

Precondition:: this->cols() == 1
Precondition:: control_points()[0].head(n) must be a valid operation.

FinalValue(self: pydrake.trajectories.BsplineTrajectory) → numpy.ndarray[numpy.float64[m, n]]: Returns this->value(this->end_time())

InitialValue(self: pydrake.trajectories.BsplineTrajectory) → numpy.ndarray[numpy.float64[m, n]]: Returns this->value(this->start_time())

InsertKnots(self: pydrake.trajectories.BsplineTrajectory, additional_knots: list[float]) → None

Adds new knots at the specified additional_knots without changing the behavior of the trajectory. The basis and control points of the trajectory are adjusted such that it produces the same value for any valid time before and after this method is called. The resulting trajectory is guaranteed to have the same level of continuity as the original, even if knot values are duplicated. Note that additional_knots need not be sorted.

Precondition:: start_time() <= t <= end_time() for all t in additional_knots

num_control_points(self: pydrake.trajectories.BsplineTrajectory) → int: Returns the number of control points in this curve.

template pydrake.trajectories.BsplineTrajectory_: Instantiations: BsplineTrajectory_[float], BsplineTrajectory_[AutoDiffXd], BsplineTrajectory_[Expression]

class pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]

Bases: pydrake.trajectories.Trajectory_[AutoDiffXd]

Represents a B-spline curve using a given basis with ordered control_points such that each control point is a matrix in ℝʳᵒʷˢ ˣ ᶜᵒˡˢ.

See also

math::BsplineBasis

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]) -> None
__init__(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd], basis: pydrake.math.BsplineBasis_[AutoDiffXd], control_points: list[list[pydrake.autodiffutils.AutoDiffXd]]) -> None

Constructs a B-spline trajectory with the given basis and control_points.

Precondition:: control_points.size() == basis.num_basis_functions()

__init__(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd], basis: pydrake.math.BsplineBasis_[AutoDiffXd], control_points: list[numpy.ndarray[object[m, n]]]) -> None

Constructs a B-spline trajectory with the given basis and control_points.

Precondition:: control_points.size() == basis.num_basis_functions()

basis(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]) → pydrake.math.BsplineBasis_[AutoDiffXd]: Returns the basis of this curve.

Clone(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]) → pydrake.trajectories.Trajectory_[AutoDiffXd]

control_points(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]) → list[numpy.ndarray[object[m, n]]]: Returns the control points of this curve.

CopyBlock(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd], start_row: int, start_col: int, block_rows: int, block_cols: int) → pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]: Returns a new BsplineTrajectory that uses the same basis as this, and whose control points are the result of calling point.block(start_row, start_col, block_rows, block_cols) on each point in this->control_points().

CopyHead(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd], n: int) → pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]

Returns a new BsplineTrajectory that uses the same basis as this, and whose control points are the result of calling point.head(n) on each point in this->control_points().

Precondition:: this->cols() == 1
Precondition:: control_points()[0].head(n) must be a valid operation.

FinalValue(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]) → numpy.ndarray[object[m, n]]: Returns this->value(this->end_time())

InitialValue(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]) → numpy.ndarray[object[m, n]]: Returns this->value(this->start_time())

InsertKnots(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd], additional_knots: list[pydrake.autodiffutils.AutoDiffXd]) → None

Adds new knots at the specified additional_knots without changing the behavior of the trajectory. The basis and control points of the trajectory are adjusted such that it produces the same value for any valid time before and after this method is called. The resulting trajectory is guaranteed to have the same level of continuity as the original, even if knot values are duplicated. Note that additional_knots need not be sorted.

Precondition:: start_time() <= t <= end_time() for all t in additional_knots

num_control_points(self: pydrake.trajectories.BsplineTrajectory_[AutoDiffXd]) → int: Returns the number of control points in this curve.

class pydrake.trajectories.BsplineTrajectory_[Expression]

Bases: pydrake.trajectories.Trajectory_[Expression]

Represents a B-spline curve using a given basis with ordered control_points such that each control point is a matrix in ℝʳᵒʷˢ ˣ ᶜᵒˡˢ.

See also

math::BsplineBasis

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.BsplineTrajectory_[Expression]) -> None
__init__(self: pydrake.trajectories.BsplineTrajectory_[Expression], basis: pydrake.math.BsplineBasis_[Expression], control_points: list[list[pydrake.symbolic.Expression]]) -> None

Constructs a B-spline trajectory with the given basis and control_points.

Precondition:: control_points.size() == basis.num_basis_functions()

__init__(self: pydrake.trajectories.BsplineTrajectory_[Expression], basis: pydrake.math.BsplineBasis_[Expression], control_points: list[numpy.ndarray[object[m, n]]]) -> None

Constructs a B-spline trajectory with the given basis and control_points.

Precondition:: control_points.size() == basis.num_basis_functions()

basis(self: pydrake.trajectories.BsplineTrajectory_[Expression]) → pydrake.math.BsplineBasis_[Expression]: Returns the basis of this curve.

Clone(self: pydrake.trajectories.BsplineTrajectory_[Expression]) → pydrake.trajectories.Trajectory_[Expression]

control_points(self: pydrake.trajectories.BsplineTrajectory_[Expression]) → list[numpy.ndarray[object[m, n]]]: Returns the control points of this curve.

CopyBlock(self: pydrake.trajectories.BsplineTrajectory_[Expression], start_row: int, start_col: int, block_rows: int, block_cols: int) → pydrake.trajectories.BsplineTrajectory_[Expression]: Returns a new BsplineTrajectory that uses the same basis as this, and whose control points are the result of calling point.block(start_row, start_col, block_rows, block_cols) on each point in this->control_points().

CopyHead(self: pydrake.trajectories.BsplineTrajectory_[Expression], n: int) → pydrake.trajectories.BsplineTrajectory_[Expression]

Returns a new BsplineTrajectory that uses the same basis as this, and whose control points are the result of calling point.head(n) on each point in this->control_points().

Precondition:: this->cols() == 1
Precondition:: control_points()[0].head(n) must be a valid operation.

FinalValue(self: pydrake.trajectories.BsplineTrajectory_[Expression]) → numpy.ndarray[object[m, n]]: Returns this->value(this->end_time())

InitialValue(self: pydrake.trajectories.BsplineTrajectory_[Expression]) → numpy.ndarray[object[m, n]]: Returns this->value(this->start_time())

InsertKnots(self: pydrake.trajectories.BsplineTrajectory_[Expression], additional_knots: list[pydrake.symbolic.Expression]) → None

Adds new knots at the specified additional_knots without changing the behavior of the trajectory. The basis and control points of the trajectory are adjusted such that it produces the same value for any valid time before and after this method is called. The resulting trajectory is guaranteed to have the same level of continuity as the original, even if knot values are duplicated. Note that additional_knots need not be sorted.

Precondition:: start_time() <= t <= end_time() for all t in additional_knots

num_control_points(self: pydrake.trajectories.BsplineTrajectory_[Expression]) → int: Returns the number of control points in this curve.

class pydrake.trajectories.CompositeTrajectory

Bases: pydrake.trajectories.PiecewiseTrajectory

A “composite trajectory” is a series of trajectories joined end to end where the end time of one trajectory coincides with the starting time of the next.

See also PiecewisePolynomial::ConcatenateInTime(), which might be preferred if all of the segments are PiecewisePolynomial.

Note

This class is templated; see CompositeTrajectory_ for the list of instantiations.

__init__(self: pydrake.trajectories.CompositeTrajectory, segments: list[pydrake.trajectories.Trajectory]) → None

Constructs a composite trajectory from a list of Trajectories.

Precondition:: ∀i, segments[i].get() != nullptr.
Precondition:: ∀i, segments[i+1].start_time() == segments[i].end_time().
Precondition:: ∀i, segments[i].rows() == segments[0].rows() and segments[i].cols() == segments[0].cols()`.

static AlignAndConcatenate(segments: list[pydrake.trajectories.Trajectory]) → pydrake.trajectories.CompositeTrajectory

Constructs a composite trajectory from a list of trajectories whose start and end times may not coincide, by translating their start and end times.

Precondition:: ∀i, segments[i].get() != nullptr.
Precondition:: ∀i, segments[i].rows() == segments[0].rows() and segments[i].cols() == segments[0].cols()`.

segment(self: pydrake.trajectories.CompositeTrajectory, segment_index: int) → pydrake.trajectories.Trajectory: Returns a reference to the segment_index trajectory.

template pydrake.trajectories.CompositeTrajectory_: Instantiations: CompositeTrajectory_[float], CompositeTrajectory_[AutoDiffXd], CompositeTrajectory_[Expression]

class pydrake.trajectories.CompositeTrajectory_[AutoDiffXd]

Bases: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd]

A “composite trajectory” is a series of trajectories joined end to end where the end time of one trajectory coincides with the starting time of the next.

See also PiecewisePolynomial::ConcatenateInTime(), which might be preferred if all of the segments are PiecewisePolynomial.

__init__(self: pydrake.trajectories.CompositeTrajectory_[AutoDiffXd], segments: list[pydrake.trajectories.Trajectory_[AutoDiffXd]]) → None

Constructs a composite trajectory from a list of Trajectories.

Precondition:: ∀i, segments[i].get() != nullptr.
Precondition:: ∀i, segments[i+1].start_time() == segments[i].end_time().
Precondition:: ∀i, segments[i].rows() == segments[0].rows() and segments[i].cols() == segments[0].cols()`.

static AlignAndConcatenate(segments: list[pydrake.trajectories.Trajectory_[AutoDiffXd]]) → pydrake.trajectories.CompositeTrajectory_[AutoDiffXd]

Constructs a composite trajectory from a list of trajectories whose start and end times may not coincide, by translating their start and end times.

Precondition:: ∀i, segments[i].get() != nullptr.
Precondition:: ∀i, segments[i].rows() == segments[0].rows() and segments[i].cols() == segments[0].cols()`.

segment(self: pydrake.trajectories.CompositeTrajectory_[AutoDiffXd], segment_index: int) → pydrake.trajectories.Trajectory_[AutoDiffXd]: Returns a reference to the segment_index trajectory.

class pydrake.trajectories.CompositeTrajectory_[Expression]

Bases: pydrake.trajectories.PiecewiseTrajectory_[Expression]

A “composite trajectory” is a series of trajectories joined end to end where the end time of one trajectory coincides with the starting time of the next.

See also PiecewisePolynomial::ConcatenateInTime(), which might be preferred if all of the segments are PiecewisePolynomial.

__init__(self: pydrake.trajectories.CompositeTrajectory_[Expression], segments: list[pydrake.trajectories.Trajectory_[Expression]]) → None

Constructs a composite trajectory from a list of Trajectories.

Precondition:: ∀i, segments[i].get() != nullptr.
Precondition:: ∀i, segments[i+1].start_time() == segments[i].end_time().
Precondition:: ∀i, segments[i].rows() == segments[0].rows() and segments[i].cols() == segments[0].cols()`.

static AlignAndConcatenate(segments: list[pydrake.trajectories.Trajectory_[Expression]]) → pydrake.trajectories.CompositeTrajectory_[Expression]

Constructs a composite trajectory from a list of trajectories whose start and end times may not coincide, by translating their start and end times.

Precondition:: ∀i, segments[i].get() != nullptr.
Precondition:: ∀i, segments[i].rows() == segments[0].rows() and segments[i].cols() == segments[0].cols()`.

segment(self: pydrake.trajectories.CompositeTrajectory_[Expression], segment_index: int) → pydrake.trajectories.Trajectory_[Expression]: Returns a reference to the segment_index trajectory.

class pydrake.trajectories.DerivativeTrajectory

Bases: pydrake.trajectories.Trajectory

Trajectory objects provide derivatives by implementing DoEvalDerivative and DoMakeDerivative. DoEvalDerivative evaluates the derivative value at a point in time. DoMakeDerivative returns a new Trajectory object which represents the derivative.

In some cases, it is easy to implement DoEvalDerivative, but difficult or inefficient to implement DoMakeDerivative natively. And it may be just as efficient to use DoEvalDerivative even in repeated evaluations of the derivative. The DerivativeTrajectory class helps with this case – given a nominal Trajectory, it provides a Trajectory interface that calls nominal.EvalDerivative() to implement Trajectory::value().

Note

This class is templated; see DerivativeTrajectory_ for the list of instantiations.

__init__(self: pydrake.trajectories.DerivativeTrajectory, nominal: pydrake.trajectories.Trajectory, derivative_order: int = 1) → None

Creates a DerivativeTrajectory representing the derivative_order derivatives of nominal. This constructor makes a Clone() of nominal and does not hold on to the reference.

Raises

RuntimeError if !nominal.has_derivative() –
RuntimeError if derivative_order < 0. –

Clone(self: pydrake.trajectories.DerivativeTrajectory) → pydrake.trajectories.Trajectory

template pydrake.trajectories.DerivativeTrajectory_: Instantiations: DerivativeTrajectory_[float], DerivativeTrajectory_[AutoDiffXd], DerivativeTrajectory_[Expression]

class pydrake.trajectories.DerivativeTrajectory_[AutoDiffXd]

Bases: pydrake.trajectories.Trajectory_[AutoDiffXd]

Trajectory objects provide derivatives by implementing DoEvalDerivative and DoMakeDerivative. DoEvalDerivative evaluates the derivative value at a point in time. DoMakeDerivative returns a new Trajectory object which represents the derivative.

In some cases, it is easy to implement DoEvalDerivative, but difficult or inefficient to implement DoMakeDerivative natively. And it may be just as efficient to use DoEvalDerivative even in repeated evaluations of the derivative. The DerivativeTrajectory class helps with this case – given a nominal Trajectory, it provides a Trajectory interface that calls nominal.EvalDerivative() to implement Trajectory::value().

__init__(self: pydrake.trajectories.DerivativeTrajectory_[AutoDiffXd], nominal: pydrake.trajectories.Trajectory_[AutoDiffXd], derivative_order: int = 1) → None

Creates a DerivativeTrajectory representing the derivative_order derivatives of nominal. This constructor makes a Clone() of nominal and does not hold on to the reference.

Raises

RuntimeError if !nominal.has_derivative() –
RuntimeError if derivative_order < 0. –

Clone(self: pydrake.trajectories.DerivativeTrajectory_[AutoDiffXd]) → pydrake.trajectories.Trajectory_[AutoDiffXd]

class pydrake.trajectories.DerivativeTrajectory_[Expression]

Bases: pydrake.trajectories.Trajectory_[Expression]

Trajectory objects provide derivatives by implementing DoEvalDerivative and DoMakeDerivative. DoEvalDerivative evaluates the derivative value at a point in time. DoMakeDerivative returns a new Trajectory object which represents the derivative.

In some cases, it is easy to implement DoEvalDerivative, but difficult or inefficient to implement DoMakeDerivative natively. And it may be just as efficient to use DoEvalDerivative even in repeated evaluations of the derivative. The DerivativeTrajectory class helps with this case – given a nominal Trajectory, it provides a Trajectory interface that calls nominal.EvalDerivative() to implement Trajectory::value().

__init__(self: pydrake.trajectories.DerivativeTrajectory_[Expression], nominal: pydrake.trajectories.Trajectory_[Expression], derivative_order: int = 1) → None

Creates a DerivativeTrajectory representing the derivative_order derivatives of nominal. This constructor makes a Clone() of nominal and does not hold on to the reference.

Raises

RuntimeError if !nominal.has_derivative() –
RuntimeError if derivative_order < 0. –

Clone(self: pydrake.trajectories.DerivativeTrajectory_[Expression]) → pydrake.trajectories.Trajectory_[Expression]

class pydrake.trajectories.ExponentialPlusPiecewisePolynomial

Bases: pydrake.trajectories.PiecewiseTrajectory

Represents a piecewise-trajectory with piece \(j\) given by:

\[x(t) = K e^{A (t - t_j)} \alpha_j + \sum_{i=0}^k \beta_{j,i}(t-t_j)^i,\]

where \(k\) is the order of the piecewise_polynomial_part and \(t_j\) is the start time of the \(j\)-th segment.

This particular form can represent the solution to a linear dynamical system driven by a piecewise-polynomial input:

\[\dot{x}(t) = A x(t) + B u(t),\]

where the input \(u(t)\) is a piecewise-polynomial function of time. See [1] for details and a motivating use case.

[1] R. Tedrake, S. Kuindersma, R. Deits and K. Miura, “A closed-form solution for real-time ZMP gait generation and feedback stabilization,” 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids), Seoul, 2015, pp. 936-940.

Note

This class is templated; see ExponentialPlusPiecewisePolynomial_ for the list of instantiations.

__init__(self: pydrake.trajectories.ExponentialPlusPiecewisePolynomial, K: numpy.ndarray[numpy.float64[m, n]], A: numpy.ndarray[numpy.float64[m, n]], alpha: numpy.ndarray[numpy.float64[m, n]], piecewise_polynomial_part: pydrake.trajectories.PiecewisePolynomial) → None

Clone(self: pydrake.trajectories.ExponentialPlusPiecewisePolynomial) → pydrake.trajectories.Trajectory

shiftRight(self: pydrake.trajectories.ExponentialPlusPiecewisePolynomial, offset: float) → None

template pydrake.trajectories.ExponentialPlusPiecewisePolynomial_: Instantiations: ExponentialPlusPiecewisePolynomial_[float]

class pydrake.trajectories.FunctionHandleTrajectory

Bases: pydrake.trajectories.Trajectory

FunctionHandleTrajectory takes a function, value = f(t), and provides a Trajectory interface.

Note

This class is templated; see FunctionHandleTrajectory_ for the list of instantiations.

__init__(self: pydrake.trajectories.FunctionHandleTrajectory, func: Callable[[float], numpy.ndarray[numpy.float64[m, n]]], rows: int, cols: int = 1, start_time: float = - inf, end_time: float = inf) → None

Creates the FunctionHandleTrajectory.

Parameter func:: The function to be used to evaluate the trajectory.
Parameter rows:: The number of rows in the output of the function.
Parameter cols:: The number of columns in the output of the function.
Parameter start_time:: The start time of the trajectory.
Parameter end_time:: The end time of the trajectory.

Raises

RuntimeError if func == nullptr, rows < 0, cols < 0, start_time > –
end_time, or if the function returns a matrix of the wrong size. –

Clone(self: pydrake.trajectories.FunctionHandleTrajectory) → pydrake.trajectories.Trajectory

template pydrake.trajectories.FunctionHandleTrajectory_: Instantiations: FunctionHandleTrajectory_[float], FunctionHandleTrajectory_[AutoDiffXd], FunctionHandleTrajectory_[Expression]

class pydrake.trajectories.FunctionHandleTrajectory_[AutoDiffXd]

Bases: pydrake.trajectories.Trajectory_[AutoDiffXd]

FunctionHandleTrajectory takes a function, value = f(t), and provides a Trajectory interface.

__init__(self: pydrake.trajectories.FunctionHandleTrajectory_[AutoDiffXd], func: Callable[[pydrake.autodiffutils.AutoDiffXd], numpy.ndarray[object[m, n]]], rows: int, cols: int = 1, start_time: float = - inf, end_time: float = inf) → None

Creates the FunctionHandleTrajectory.

Parameter func:: The function to be used to evaluate the trajectory.
Parameter rows:: The number of rows in the output of the function.
Parameter cols:: The number of columns in the output of the function.
Parameter start_time:: The start time of the trajectory.
Parameter end_time:: The end time of the trajectory.

Raises

RuntimeError if func == nullptr, rows < 0, cols < 0, start_time > –
end_time, or if the function returns a matrix of the wrong size. –

Clone(self: pydrake.trajectories.FunctionHandleTrajectory_[AutoDiffXd]) → pydrake.trajectories.Trajectory_[AutoDiffXd]

class pydrake.trajectories.FunctionHandleTrajectory_[Expression]

Bases: pydrake.trajectories.Trajectory_[Expression]

FunctionHandleTrajectory takes a function, value = f(t), and provides a Trajectory interface.

__init__(self: pydrake.trajectories.FunctionHandleTrajectory_[Expression], func: Callable[[pydrake.symbolic.Expression], numpy.ndarray[object[m, n]]], rows: int, cols: int = 1, start_time: float = - inf, end_time: float = inf) → None

Creates the FunctionHandleTrajectory.

Parameter func:: The function to be used to evaluate the trajectory.
Parameter rows:: The number of rows in the output of the function.
Parameter cols:: The number of columns in the output of the function.
Parameter start_time:: The start time of the trajectory.
Parameter end_time:: The end time of the trajectory.

Raises

RuntimeError if func == nullptr, rows < 0, cols < 0, start_time > –
end_time, or if the function returns a matrix of the wrong size. –

Clone(self: pydrake.trajectories.FunctionHandleTrajectory_[Expression]) → pydrake.trajectories.Trajectory_[Expression]

class pydrake.trajectories.PathParameterizedTrajectory

Bases: pydrake.trajectories.Trajectory

A trajectory defined by a path and timing trajectory.

Using a path of form r(s) and a time_scaling of the form s(t), a full trajectory of form q(t) = r(s(t)) is modeled.

Note

This class is templated; see PathParameterizedTrajectory_ for the list of instantiations.

__init__(self: pydrake.trajectories.PathParameterizedTrajectory, path: pydrake.trajectories.Trajectory, time_scaling: pydrake.trajectories.Trajectory) → None

Constructs a trajectory with the given path and time_scaling.

Precondition:: time_scaling.rows() == time_scaling.cols() == 1

Clone(self: pydrake.trajectories.PathParameterizedTrajectory) → pydrake.trajectories.Trajectory

path(self: pydrake.trajectories.PathParameterizedTrajectory) → pydrake.trajectories.Trajectory: Returns the path of this trajectory.

time_scaling(self: pydrake.trajectories.PathParameterizedTrajectory) → pydrake.trajectories.Trajectory: Returns the time_scaling of this trajectory.

template pydrake.trajectories.PathParameterizedTrajectory_: Instantiations: PathParameterizedTrajectory_[float], PathParameterizedTrajectory_[AutoDiffXd], PathParameterizedTrajectory_[Expression]

class pydrake.trajectories.PathParameterizedTrajectory_[AutoDiffXd]

Bases: pydrake.trajectories.Trajectory_[AutoDiffXd]

A trajectory defined by a path and timing trajectory.

Using a path of form r(s) and a time_scaling of the form s(t), a full trajectory of form q(t) = r(s(t)) is modeled.

__init__(self: pydrake.trajectories.PathParameterizedTrajectory_[AutoDiffXd], path: pydrake.trajectories.Trajectory_[AutoDiffXd], time_scaling: pydrake.trajectories.Trajectory_[AutoDiffXd]) → None

Constructs a trajectory with the given path and time_scaling.

Precondition:: time_scaling.rows() == time_scaling.cols() == 1

Clone(self: pydrake.trajectories.PathParameterizedTrajectory_[AutoDiffXd]) → pydrake.trajectories.Trajectory_[AutoDiffXd]

path(self: pydrake.trajectories.PathParameterizedTrajectory_[AutoDiffXd]) → pydrake.trajectories.Trajectory_[AutoDiffXd]: Returns the path of this trajectory.

time_scaling(self: pydrake.trajectories.PathParameterizedTrajectory_[AutoDiffXd]) → pydrake.trajectories.Trajectory_[AutoDiffXd]: Returns the time_scaling of this trajectory.

class pydrake.trajectories.PathParameterizedTrajectory_[Expression]

Bases: pydrake.trajectories.Trajectory_[Expression]

A trajectory defined by a path and timing trajectory.

Using a path of form r(s) and a time_scaling of the form s(t), a full trajectory of form q(t) = r(s(t)) is modeled.

__init__(self: pydrake.trajectories.PathParameterizedTrajectory_[Expression], path: pydrake.trajectories.Trajectory_[Expression], time_scaling: pydrake.trajectories.Trajectory_[Expression]) → None

Constructs a trajectory with the given path and time_scaling.

Precondition:: time_scaling.rows() == time_scaling.cols() == 1

Clone(self: pydrake.trajectories.PathParameterizedTrajectory_[Expression]) → pydrake.trajectories.Trajectory_[Expression]

path(self: pydrake.trajectories.PathParameterizedTrajectory_[Expression]) → pydrake.trajectories.Trajectory_[Expression]: Returns the path of this trajectory.

time_scaling(self: pydrake.trajectories.PathParameterizedTrajectory_[Expression]) → pydrake.trajectories.Trajectory_[Expression]: Returns the time_scaling of this trajectory.

class pydrake.trajectories.PiecewisePolynomial

Bases: pydrake.trajectories.PiecewiseTrajectory

A scalar multi-variate piecewise polynomial.

PiecewisePolynomial represents a list of contiguous segments in a scalar independent variable (typically corresponding to time) with Polynomials defined at each segment. We call the output from evaluating the PiecewisePolynomial at the scalar independent variable “the output”, and that output can be either a Eigen MatrixX<T> (if evaluated using value()) or a scalar (if evaluated using scalar_value()).

An example of a piecewise polynomial is a function of m segments in time, where a different polynomial is defined for each segment. For a specific example, consider the absolute value function over the interval [-1, 1]. We can define a PiecewisePolynomial over this interval using breaks at t = { -1.0, 0.0, 1.0 }, and “samples” of abs(t).

Click to expand C++ code...

// Construct the PiecewisePolynomial.
const std::vector<double> breaks = { -1.0, 0.0, 1.0 };
std::vector<Eigen::MatrixXd> samples(3);
for (int i = 0; i < static_cast<int>(breaks.size()); ++i) {
  samples[i].resize(1, 1);
  samples[i](0, 0) = std::abs(breaks[i]);
}
const auto pp =
     PiecewisePolynomial<double>::FirstOrderHold(breaks, samples);
const int row = 0, col = 0;

// Evaluate the PiecewisePolynomial at some values.
std::cout << pp.value(-.5)(row, col) << std::endl;    // Outputs 0.5.
std::cout << pp.value(0.0)(row, col) << std::endl;    // Outputs 0.0;

// Show how we can evaluate the first derivative (outputs -1.0).
std::cout << pp.derivative(1).value(-.5)(row, col) << std::endl;

A note on terminology. For piecewise-polynomial interpolation, we use breaks to indicate the scalar (e.g. times) which form the boundary of each segment. We use samples to indicate the function value at the breaks, e.g. p(breaks[i]) = samples[i]. The term knot should be reserved for the “(x,y)” coordinate, here knot[i] = (breaks[i], samples[i]), though it is used inconsistently in the interpolation literature (sometimes for breaks, sometimes for samples), so we try to mostly avoid it here.

PiecewisePolynomial objects can be added, subtracted, and multiplied. They cannot be divided because Polynomials are not closed under division.

Warning

PiecewisePolynomial silently clips input evaluations outside of the range defined by the breaks. So pp.value(-2.0, row, col) in the example above would evaluate to -1.0. See value().

Note

This class is templated; see PiecewisePolynomial_ for the list of instantiations.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.PiecewisePolynomial) -> None

Constructs an empty piecewise polynomial.

__init__(self: pydrake.trajectories.PiecewisePolynomial, arg0: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) -> None

Single segment, constant value constructor over the interval [-∞, ∞]. The constructed PiecewisePolynomial will return constant_value at every evaluated point (i.e., value(t) = constant_value ∀t ∈ [-∞, ∞]).

__init__(self: pydrake.trajectories.PiecewisePolynomial, arg0: list[numpy.ndarray[object[m, n]]], arg1: list[float]) -> None

Constructs a PiecewisePolynomial using matrix-output Polynomials defined over each segment.

Precondition:: polynomials.size() == breaks.size() - 1

__init__(self: pydrake.trajectories.PiecewisePolynomial, arg0: list[pydrake.polynomial.Polynomial], arg1: list[float]) -> None

Constructs a PiecewisePolynomial using scalar-output Polynomials defined over each segment.

Precondition:: polynomials.size() == breaks.size() - 1

AppendCubicHermiteSegment(self: pydrake.trajectories.PiecewisePolynomial, time: float, sample: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], sample_dot: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → None

The CubicHermite spline construction has a nice property of being incremental (each segment can be solved independently). Given a new sample and it’s derivative, this method adds one segment to the end of this where the start sample and derivative are taken as the value and derivative at the final break of this.

Precondition:: this is not empty()
Precondition:: time > end_time()
Precondition:: sample and sample_dot must have size rows() x cols().

AppendFirstOrderSegment(self: pydrake.trajectories.PiecewisePolynomial, time: float, sample: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → None: Given a new sample, this method adds one segment to the end of this using a first-order hold, where the start sample is taken as the value at the final break of this.

Block(self: pydrake.trajectories.PiecewisePolynomial, start_row: int, start_col: int, block_rows: int, block_cols: int) → pydrake.trajectories.PiecewisePolynomial

Extracts a trajectory representing a block of size (block_rows, block_cols) starting at (start_row, start_col) from the PiecewisePolynomial.

Returns: a PiecewisePolynomial such that ret.value(t) = this.value(t).block(i,j,p,q);

Clone(self: pydrake.trajectories.PiecewisePolynomial) → pydrake.trajectories.Trajectory

ConcatenateInTime(self: pydrake.trajectories.PiecewisePolynomial, other: pydrake.trajectories.PiecewisePolynomial) → None

Concatenates other to the end of this.

Warning

The resulting PiecewisePolynomial will only be continuous to the degree that the first Polynomial of other is continuous with the last Polynomial of this. See warning about evaluating discontinuous derivatives at breaks in derivative().

Parameter other:: PiecewisePolynomial instance to concatenate.

Raises

RuntimeError if trajectories' dimensions do not match each other –
(either rows() or cols() does not match between this and –
other`) –
RuntimeError if this->end_time() and other->start_time() –
are not within PiecewiseTrajectory<T>::kEpsilonTime from each –
other. –

static CubicHermite(*args, **kwargs)

Overloaded function.

CubicHermite(breaks: list[float], samples: list[list[float]], samples_dot: list[list[float]]) -> pydrake.trajectories.PiecewisePolynomial

Version of CubicHermite(breaks, samples, samples_dot) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Corresponding columns of samples and samples_dot are used as the sample point and independent variable derivative, respectively.

Precondition:: samples.cols() == samples_dot.cols() == breaks.size().

CubicHermite(breaks: list[float], samples: list[numpy.ndarray[numpy.float64[m, n]]], samples_dot: list[numpy.ndarray[numpy.float64[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial

Constructs a third order PiecewisePolynomial using matrix samples and derivatives of samples (samples_dot); each matrix element of samples_dot represents the derivative with respect to the independent variable (e.g., the time derivative) of the corresponding entry in samples. Each segment is fully specified by samples and sample_dot at both ends. Second derivatives are not continuous.

static CubicShapePreserving(*args, **kwargs)

Overloaded function.

CubicShapePreserving(breaks: list[float], samples: list[list[float]], zero_end_point_derivatives: bool = False) -> pydrake.trajectories.PiecewisePolynomial

Version of CubicShapePreserving(breaks, samples, zero_end_point_derivatives) that uses vector samples and Eigen VectorXd and MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size().

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

CubicShapePreserving(breaks: list[float], samples: list[numpy.ndarray[numpy.float64[m, n]]], zero_end_point_derivatives: bool = False) -> pydrake.trajectories.PiecewisePolynomial

Constructs a third order PiecewisePolynomial using vector samples, where each column of samples represents a sample point. First derivatives are chosen to be “shape preserving”, i.e. if samples is monotonic within some interval, the interpolated data will also be monotonic. The second derivative is not guaranteed to be smooth across the entire spline.

MATLAB calls this method “pchip” (short for “Piecewise Cubic Hermite Interpolating Polynomial”), and provides a nice description in their documentation. http://home.uchicago.edu/~sctchoi/courses/cs138/interp.pdf is also a good reference.

If zero_end_point_derivatives is False, the first and last first derivative is chosen using a non-centered, shape-preserving three-point formulae. See equation (2.10) in the following reference for more details. http://www.mi.sanu.ac.rs/~gvm/radovi/mon.pdf If zero_end_point_derivatives is True, they are set to zeros.

If zero_end_point_derivatives is False, breaks and samples must have at least 3 elements for the algorithm to determine the first derivatives.

If zero_end_point_derivatives is True, breaks and samples may have 2 or more elements. For the 2 elements case, the result is equivalent to computing a cubic polynomial whose values are given by samples, and derivatives set to zero.

Raises

RuntimeError if –
- breaks has length smaller than 3 and
zero_end_point_derivatives` is False, - breaks has lengt –
smaller than 2 and zero_end_point_derivatives is true. –
RuntimeError under the conditions specified under –
coefficient_construction_methods. –

static CubicWithContinuousSecondDerivatives(*args, **kwargs)

Overloaded function.

CubicWithContinuousSecondDerivatives(breaks: list[float], samples: list[list[float]], sample_dot_at_start: numpy.ndarray[numpy.float64[m, n]], sample_dot_at_end: numpy.ndarray[numpy.float64[m, n]]) -> pydrake.trajectories.PiecewisePolynomial

Version of CubicWithContinuousSecondDerivatives() that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size().

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

CubicWithContinuousSecondDerivatives(breaks: list[float], samples: list[numpy.ndarray[numpy.float64[m, n]]], sample_dot_at_start: numpy.ndarray[numpy.float64[m, n]], sample_dot_at_end: numpy.ndarray[numpy.float64[m, n]]) -> pydrake.trajectories.PiecewisePolynomial

Constructs a third order PiecewisePolynomial using matrix samples. The PiecewisePolynomial is constructed such that the interior segments have the same value, first and second derivatives at breaks. sample_dot_at_start and sample_dot_at_end are used for the first and last first derivatives.

Raises

RuntimeError if sample_dot_at_start or sample_dot_at_end –
and samples have inconsistent dimensions. –
RuntimeError under the conditions specified under –
coefficient_construction_methods. –

CubicWithContinuousSecondDerivatives(breaks: list[float], samples: list[list[float]], periodic_end_condition: bool = False) -> pydrake.trajectories.PiecewisePolynomial

Version of CubicWithContinuousSecondDerivatives(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size().

CubicWithContinuousSecondDerivatives(breaks: list[float], samples: list[numpy.ndarray[numpy.float64[m, n]]], periodic_end: bool) -> pydrake.trajectories.PiecewisePolynomial

Constructs a third order PiecewisePolynomial using matrix samples. The PiecewisePolynomial is constructed such that the interior segments have the same value, first and second derivatives at breaks. If periodic_end_condition is False (default), then the “Not-a-sample” end condition is used here, which means the third derivatives are continuous for the first two and last two segments. If periodic_end_condition is True, then the first and second derivatives between the end of the last segment and the beginning of the first segment will be continuous. Note that the periodic end condition does not require the first and last sample to be collocated, nor does it add an additional sample to connect the first and last segments. Only first and second derivative continuity is enforced. See https://en.wikipedia.org/wiki/Spline_interpolation and https://web.archive.org/web/20140125011904/https://www.math.uh.edu/~jingqiu/math4364/spline.pdf for more about cubic splines and their end conditions. The MATLAB docs for methods “spline” and “csape” are also good references.

Precondition:: breaks and samples must have at least 3 elements. If periodic_end_condition is True, then for two samples, it would produce a straight line (use FirstOrderHold for this instead), and if periodic_end_condition is False the problem is ill-defined.

derivative(self: pydrake.trajectories.PiecewisePolynomial, derivative_order: int = 1) → pydrake.trajectories.PiecewisePolynomial

Returns a PiecewisePolynomial where each segment is the specified derivative of the corresponding segment in this. Any rules or limitations of Polynomial::derivative() also apply to this function.

Derivatives evaluated at non-differentiable points return the value at the left hand side of the interval.

Parameter derivative_order:: The order of the derivative, namely, if derivative_order = n, the n’th derivative of the polynomial will be returned.

Warning

In the event of discontinuous derivatives evaluated at breaks, it is not defined which polynomial (i.e., to the left or right of the break) will be the one that is evaluated at the break.

static FirstOrderHold(*args, **kwargs)

Overloaded function.

FirstOrderHold(breaks: list[float], samples: list[list[float]]) -> pydrake.trajectories.PiecewisePolynomial

Version of FirstOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size()

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

FirstOrderHold(breaks: list[float], samples: list[numpy.ndarray[numpy.float64[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial

Constructs a piecewise linear PiecewisePolynomial using matrix samples.

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

getPolynomial(self: pydrake.trajectories.PiecewisePolynomial, segment_index: int, row: int = 0, col: int = 0) → pydrake.polynomial.Polynomial: Gets the Polynomial with the given matrix row and column index that corresponds to the given segment index. Equivalent to getPolynomialMatrix(segment_index)(row, col).

Note

Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

getPolynomialMatrix(self: pydrake.trajectories.PiecewisePolynomial, segment_index: int) → numpy.ndarray[object[m, n]]: Gets the matrix of Polynomials corresponding to the given segment index.

Warning

segment_index is not checked for validity.

getSegmentPolynomialDegree(self: pydrake.trajectories.PiecewisePolynomial, segment_index: int, row: int = 0, col: int = 0) → int: Gets the degree of the Polynomial with the given matrix row and column index that corresponds to the given segment index. Equivalent to getPolynomial(segment_index, row, col).GetDegree().

isApprox(self: pydrake.trajectories.PiecewisePolynomial, other: pydrake.trajectories.PiecewisePolynomial, tol: float, tol_type: pydrake.common.ToleranceType = <ToleranceType.kRelative: 1>) → bool: Checks whether a PiecewisePolynomial is approximately equal to this one by calling Polynomial<T>::CoefficientsAlmostEqual() on every element of every segment.

See also

Polynomial<T>::CoefficientsAlmostEqual().

static LagrangeInterpolatingPolynomial(*args, **kwargs)

Overloaded function.

LagrangeInterpolatingPolynomial(times: list[float], samples: list[list[float]]) -> pydrake.trajectories.PiecewisePolynomial

Version of LagrangeInterpolatingPolynomial(times, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == times.size().

LagrangeInterpolatingPolynomial(times: list[float], samples: list[numpy.ndarray[numpy.float64[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial

Constructs a polynomial with a single segment of the lowest possible degree that passes through all of the sample points. See “polynomial interpolation” and/or “Lagrange polynomial” on Wikipedia for more information.

Precondition:: times must be monotonically increasing.
Precondition:: samples.size() == times.size().

RemoveFinalSegment(self: pydrake.trajectories.PiecewisePolynomial) → None

Removes the final segment from the trajectory, reducing the number of segments by 1.

Precondition:: this is not empty()

Reshape(self: pydrake.trajectories.PiecewisePolynomial, rows: int, cols: int) → None

Reshapes the dimensions of the Eigen::MatrixX<T> returned by value(), EvalDerivative(), etc.

Precondition:: rows x cols must equal this.rows() * this.cols().

See also

Eigen::PlainObjectBase::resize().

ReverseTime(self: pydrake.trajectories.PiecewisePolynomial) → None: Modifies the trajectory so that pp_after(t) = pp_before(-t).

Note

The new trajectory will evaluate differently at precisely the break points if the original trajectory was discontinuous at the break points. This is because the segments are defined on the half-open intervals [breaks(i), breaks(i+1)), and the order of the breaks have been reversed.

ScaleTime(self: pydrake.trajectories.PiecewisePolynomial, scale: float) → None

Scales the time of the trajectory by non-negative scale (use ReverseTime() if you want to also negate time). The resulting polynomial evaluates to pp_after(t) = pp_before(t/scale).

As an example, `scale`=2 will result in a trajectory that is twice as long (start_time() and end_time() have both doubled).

setPolynomialMatrixBlock(self: pydrake.trajectories.PiecewisePolynomial, replacement: numpy.ndarray[object[m, n]], segment_index: int, row_start: int = 0, col_start: int = 0) → None: Replaces the specified block of the PolynomialMatrix at the given segment index.

Note

Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

Warning

This code relies upon Eigen to verify that the replacement block is not too large.

shiftRight(self: pydrake.trajectories.PiecewisePolynomial, offset: float) → None

Adds offset to all of the breaks. offset need not be a non-negative number. The resulting polynomial will evaluate to pp_after(t) = pp_before(t-offset).

As an example, `offset`=2 will result in the start_time() and end_time() being 2 seconds later.

slice(self: pydrake.trajectories.PiecewisePolynomial, start_segment_index: int, num_segments: int) → pydrake.trajectories.PiecewisePolynomial: Returns the PiecewisePolynomial comprising the num_segments segments starting at the specified start_segment_index.

Note

Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

Transpose(self: pydrake.trajectories.PiecewisePolynomial) → pydrake.trajectories.PiecewisePolynomial: Constructs a new PiecewisePolynomial for which value(t) == this.value(t).transpose().

static ZeroOrderHold(*args, **kwargs)

Overloaded function.

ZeroOrderHold(breaks: list[float], samples: list[list[float]]) -> pydrake.trajectories.PiecewisePolynomial

Version of ZeroOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd/MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size()

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

ZeroOrderHold(breaks: list[float], samples: list[numpy.ndarray[numpy.float64[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial

Constructs a piecewise constant PiecewisePolynomial using matrix samples. Note that constructing a PiecewisePolynomial requires at least two sample points, although in this case, the second sample point’s value is ignored, and only its break time is used.

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

template pydrake.trajectories.PiecewisePolynomial_: Instantiations: PiecewisePolynomial_[float], PiecewisePolynomial_[AutoDiffXd], PiecewisePolynomial_[Expression]

class pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Bases: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd]

A scalar multi-variate piecewise polynomial.

PiecewisePolynomial represents a list of contiguous segments in a scalar independent variable (typically corresponding to time) with Polynomials defined at each segment. We call the output from evaluating the PiecewisePolynomial at the scalar independent variable “the output”, and that output can be either a Eigen MatrixX<T> (if evaluated using value()) or a scalar (if evaluated using scalar_value()).

An example of a piecewise polynomial is a function of m segments in time, where a different polynomial is defined for each segment. For a specific example, consider the absolute value function over the interval [-1, 1]. We can define a PiecewisePolynomial over this interval using breaks at t = { -1.0, 0.0, 1.0 }, and “samples” of abs(t).

Click to expand C++ code...

// Construct the PiecewisePolynomial.
const std::vector<double> breaks = { -1.0, 0.0, 1.0 };
std::vector<Eigen::MatrixXd> samples(3);
for (int i = 0; i < static_cast<int>(breaks.size()); ++i) {
  samples[i].resize(1, 1);
  samples[i](0, 0) = std::abs(breaks[i]);
}
const auto pp =
     PiecewisePolynomial<double>::FirstOrderHold(breaks, samples);
const int row = 0, col = 0;

// Evaluate the PiecewisePolynomial at some values.
std::cout << pp.value(-.5)(row, col) << std::endl;    // Outputs 0.5.
std::cout << pp.value(0.0)(row, col) << std::endl;    // Outputs 0.0;

// Show how we can evaluate the first derivative (outputs -1.0).
std::cout << pp.derivative(1).value(-.5)(row, col) << std::endl;

A note on terminology. For piecewise-polynomial interpolation, we use breaks to indicate the scalar (e.g. times) which form the boundary of each segment. We use samples to indicate the function value at the breaks, e.g. p(breaks[i]) = samples[i]. The term knot should be reserved for the “(x,y)” coordinate, here knot[i] = (breaks[i], samples[i]), though it is used inconsistently in the interpolation literature (sometimes for breaks, sometimes for samples), so we try to mostly avoid it here.

PiecewisePolynomial objects can be added, subtracted, and multiplied. They cannot be divided because Polynomials are not closed under division.

Warning

PiecewisePolynomial silently clips input evaluations outside of the range defined by the breaks. So pp.value(-2.0, row, col) in the example above would evaluate to -1.0. See value().

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]) -> None

Constructs an empty piecewise polynomial.

__init__(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], arg0: numpy.ndarray[object[m, n], flags.f_contiguous]) -> None

Single segment, constant value constructor over the interval [-∞, ∞]. The constructed PiecewisePolynomial will return constant_value at every evaluated point (i.e., value(t) = constant_value ∀t ∈ [-∞, ∞]).

__init__(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], arg0: list[numpy.ndarray[object[m, n]]], arg1: list[pydrake.autodiffutils.AutoDiffXd]) -> None

Constructs a PiecewisePolynomial using matrix-output Polynomials defined over each segment.

Precondition:: polynomials.size() == breaks.size() - 1

__init__(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], arg0: list[pydrake.polynomial.Polynomial_[AutoDiffXd]], arg1: list[pydrake.autodiffutils.AutoDiffXd]) -> None

Constructs a PiecewisePolynomial using scalar-output Polynomials defined over each segment.

Precondition:: polynomials.size() == breaks.size() - 1

AppendCubicHermiteSegment(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd, sample: numpy.ndarray[object[m, n], flags.f_contiguous], sample_dot: numpy.ndarray[object[m, n], flags.f_contiguous]) → None

The CubicHermite spline construction has a nice property of being incremental (each segment can be solved independently). Given a new sample and it’s derivative, this method adds one segment to the end of this where the start sample and derivative are taken as the value and derivative at the final break of this.

Precondition:: this is not empty()
Precondition:: time > end_time()
Precondition:: sample and sample_dot must have size rows() x cols().

AppendFirstOrderSegment(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd, sample: numpy.ndarray[object[m, n], flags.f_contiguous]) → None: Given a new sample, this method adds one segment to the end of this using a first-order hold, where the start sample is taken as the value at the final break of this.

Block(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], start_row: int, start_col: int, block_rows: int, block_cols: int) → pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Extracts a trajectory representing a block of size (block_rows, block_cols) starting at (start_row, start_col) from the PiecewisePolynomial.

Returns: a PiecewisePolynomial such that ret.value(t) = this.value(t).block(i,j,p,q);

Clone(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]) → pydrake.trajectories.Trajectory_[AutoDiffXd]

ConcatenateInTime(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], other: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]) → None

Concatenates other to the end of this.

Warning

The resulting PiecewisePolynomial will only be continuous to the degree that the first Polynomial of other is continuous with the last Polynomial of this. See warning about evaluating discontinuous derivatives at breaks in derivative().

Parameter other:: PiecewisePolynomial instance to concatenate.

Raises

RuntimeError if trajectories' dimensions do not match each other –
(either rows() or cols() does not match between this and –
other`) –
RuntimeError if this->end_time() and other->start_time() –
are not within PiecewiseTrajectory<T>::kEpsilonTime from each –
other. –

static CubicHermite(*args, **kwargs)

Overloaded function.

CubicHermite(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[list[pydrake.autodiffutils.AutoDiffXd]], samples_dot: list[list[pydrake.autodiffutils.AutoDiffXd]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Version of CubicHermite(breaks, samples, samples_dot) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Corresponding columns of samples and samples_dot are used as the sample point and independent variable derivative, respectively.

Precondition:: samples.cols() == samples_dot.cols() == breaks.size().

CubicHermite(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[numpy.ndarray[object[m, n]]], samples_dot: list[numpy.ndarray[object[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Constructs a third order PiecewisePolynomial using matrix samples and derivatives of samples (samples_dot); each matrix element of samples_dot represents the derivative with respect to the independent variable (e.g., the time derivative) of the corresponding entry in samples. Each segment is fully specified by samples and sample_dot at both ends. Second derivatives are not continuous.

static CubicShapePreserving(*args, **kwargs)

Overloaded function.

CubicShapePreserving(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[list[pydrake.autodiffutils.AutoDiffXd]], zero_end_point_derivatives: bool = False) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Version of CubicShapePreserving(breaks, samples, zero_end_point_derivatives) that uses vector samples and Eigen VectorXd and MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size().

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

CubicShapePreserving(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[numpy.ndarray[object[m, n]]], zero_end_point_derivatives: bool = False) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Constructs a third order PiecewisePolynomial using vector samples, where each column of samples represents a sample point. First derivatives are chosen to be “shape preserving”, i.e. if samples is monotonic within some interval, the interpolated data will also be monotonic. The second derivative is not guaranteed to be smooth across the entire spline.

MATLAB calls this method “pchip” (short for “Piecewise Cubic Hermite Interpolating Polynomial”), and provides a nice description in their documentation. http://home.uchicago.edu/~sctchoi/courses/cs138/interp.pdf is also a good reference.

If zero_end_point_derivatives is False, the first and last first derivative is chosen using a non-centered, shape-preserving three-point formulae. See equation (2.10) in the following reference for more details. http://www.mi.sanu.ac.rs/~gvm/radovi/mon.pdf If zero_end_point_derivatives is True, they are set to zeros.

If zero_end_point_derivatives is False, breaks and samples must have at least 3 elements for the algorithm to determine the first derivatives.

If zero_end_point_derivatives is True, breaks and samples may have 2 or more elements. For the 2 elements case, the result is equivalent to computing a cubic polynomial whose values are given by samples, and derivatives set to zero.

Raises

RuntimeError if –
- breaks has length smaller than 3 and
zero_end_point_derivatives` is False, - breaks has lengt –
smaller than 2 and zero_end_point_derivatives is true. –
RuntimeError under the conditions specified under –
coefficient_construction_methods. –

static CubicWithContinuousSecondDerivatives(*args, **kwargs)

Overloaded function.

CubicWithContinuousSecondDerivatives(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[list[pydrake.autodiffutils.AutoDiffXd]], sample_dot_at_start: numpy.ndarray[object[m, n]], sample_dot_at_end: numpy.ndarray[object[m, n]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Version of CubicWithContinuousSecondDerivatives() that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size().

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

CubicWithContinuousSecondDerivatives(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[numpy.ndarray[object[m, n]]], sample_dot_at_start: numpy.ndarray[object[m, n]], sample_dot_at_end: numpy.ndarray[object[m, n]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Constructs a third order PiecewisePolynomial using matrix samples. The PiecewisePolynomial is constructed such that the interior segments have the same value, first and second derivatives at breaks. sample_dot_at_start and sample_dot_at_end are used for the first and last first derivatives.

Raises

RuntimeError if sample_dot_at_start or sample_dot_at_end –
and samples have inconsistent dimensions. –
RuntimeError under the conditions specified under –
coefficient_construction_methods. –

CubicWithContinuousSecondDerivatives(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[list[pydrake.autodiffutils.AutoDiffXd]], periodic_end_condition: bool = False) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Version of CubicWithContinuousSecondDerivatives(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size().

CubicWithContinuousSecondDerivatives(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[numpy.ndarray[object[m, n]]], periodic_end: bool) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Constructs a third order PiecewisePolynomial using matrix samples. The PiecewisePolynomial is constructed such that the interior segments have the same value, first and second derivatives at breaks. If periodic_end_condition is False (default), then the “Not-a-sample” end condition is used here, which means the third derivatives are continuous for the first two and last two segments. If periodic_end_condition is True, then the first and second derivatives between the end of the last segment and the beginning of the first segment will be continuous. Note that the periodic end condition does not require the first and last sample to be collocated, nor does it add an additional sample to connect the first and last segments. Only first and second derivative continuity is enforced. See https://en.wikipedia.org/wiki/Spline_interpolation and https://web.archive.org/web/20140125011904/https://www.math.uh.edu/~jingqiu/math4364/spline.pdf for more about cubic splines and their end conditions. The MATLAB docs for methods “spline” and “csape” are also good references.

Precondition:: breaks and samples must have at least 3 elements. If periodic_end_condition is True, then for two samples, it would produce a straight line (use FirstOrderHold for this instead), and if periodic_end_condition is False the problem is ill-defined.

derivative(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], derivative_order: int = 1) → pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Returns a PiecewisePolynomial where each segment is the specified derivative of the corresponding segment in this. Any rules or limitations of Polynomial::derivative() also apply to this function.

Derivatives evaluated at non-differentiable points return the value at the left hand side of the interval.

Parameter derivative_order:: The order of the derivative, namely, if derivative_order = n, the n’th derivative of the polynomial will be returned.

Warning

In the event of discontinuous derivatives evaluated at breaks, it is not defined which polynomial (i.e., to the left or right of the break) will be the one that is evaluated at the break.

static FirstOrderHold(*args, **kwargs)

Overloaded function.

FirstOrderHold(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[list[pydrake.autodiffutils.AutoDiffXd]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Version of FirstOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size()

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

FirstOrderHold(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[numpy.ndarray[object[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Constructs a piecewise linear PiecewisePolynomial using matrix samples.

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

getPolynomial(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], segment_index: int, row: int = 0, col: int = 0) → pydrake.polynomial.Polynomial_[AutoDiffXd]: Gets the Polynomial with the given matrix row and column index that corresponds to the given segment index. Equivalent to getPolynomialMatrix(segment_index)(row, col).

Note

Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

getPolynomialMatrix(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], segment_index: int) → numpy.ndarray[object[m, n]]: Gets the matrix of Polynomials corresponding to the given segment index.

Warning

segment_index is not checked for validity.

getSegmentPolynomialDegree(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], segment_index: int, row: int = 0, col: int = 0) → int: Gets the degree of the Polynomial with the given matrix row and column index that corresponds to the given segment index. Equivalent to getPolynomial(segment_index, row, col).GetDegree().

isApprox(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], other: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], tol: float, tol_type: pydrake.common.ToleranceType = <ToleranceType.kRelative: 1>) → bool: Checks whether a PiecewisePolynomial is approximately equal to this one by calling Polynomial<T>::CoefficientsAlmostEqual() on every element of every segment.

See also

Polynomial<T>::CoefficientsAlmostEqual().

static LagrangeInterpolatingPolynomial(*args, **kwargs)

Overloaded function.

LagrangeInterpolatingPolynomial(times: list[pydrake.autodiffutils.AutoDiffXd], samples: list[list[pydrake.autodiffutils.AutoDiffXd]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Version of LagrangeInterpolatingPolynomial(times, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == times.size().

LagrangeInterpolatingPolynomial(times: list[pydrake.autodiffutils.AutoDiffXd], samples: list[numpy.ndarray[object[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Constructs a polynomial with a single segment of the lowest possible degree that passes through all of the sample points. See “polynomial interpolation” and/or “Lagrange polynomial” on Wikipedia for more information.

Precondition:: times must be monotonically increasing.
Precondition:: samples.size() == times.size().

RemoveFinalSegment(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]) → None

Removes the final segment from the trajectory, reducing the number of segments by 1.

Precondition:: this is not empty()

Reshape(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], rows: int, cols: int) → None

Reshapes the dimensions of the Eigen::MatrixX<T> returned by value(), EvalDerivative(), etc.

Precondition:: rows x cols must equal this.rows() * this.cols().

See also

Eigen::PlainObjectBase::resize().

ReverseTime(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]) → None: Modifies the trajectory so that pp_after(t) = pp_before(-t).

Note

The new trajectory will evaluate differently at precisely the break points if the original trajectory was discontinuous at the break points. This is because the segments are defined on the half-open intervals [breaks(i), breaks(i+1)), and the order of the breaks have been reversed.

ScaleTime(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], scale: pydrake.autodiffutils.AutoDiffXd) → None

Scales the time of the trajectory by non-negative scale (use ReverseTime() if you want to also negate time). The resulting polynomial evaluates to pp_after(t) = pp_before(t/scale).

As an example, `scale`=2 will result in a trajectory that is twice as long (start_time() and end_time() have both doubled).

setPolynomialMatrixBlock(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], replacement: numpy.ndarray[object[m, n]], segment_index: int, row_start: int = 0, col_start: int = 0) → None: Replaces the specified block of the PolynomialMatrix at the given segment index.

Note

Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

Warning

This code relies upon Eigen to verify that the replacement block is not too large.

shiftRight(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], offset: pydrake.autodiffutils.AutoDiffXd) → None

Adds offset to all of the breaks. offset need not be a non-negative number. The resulting polynomial will evaluate to pp_after(t) = pp_before(t-offset).

As an example, `offset`=2 will result in the start_time() and end_time() being 2 seconds later.

slice(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], start_segment_index: int, num_segments: int) → pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]: Returns the PiecewisePolynomial comprising the num_segments segments starting at the specified start_segment_index.

Note

Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

Transpose(self: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]) → pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]: Constructs a new PiecewisePolynomial for which value(t) == this.value(t).transpose().

static ZeroOrderHold(*args, **kwargs)

Overloaded function.

ZeroOrderHold(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[list[pydrake.autodiffutils.AutoDiffXd]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Version of ZeroOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd/MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size()

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

ZeroOrderHold(breaks: list[pydrake.autodiffutils.AutoDiffXd], samples: list[numpy.ndarray[object[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]

Constructs a piecewise constant PiecewisePolynomial using matrix samples. Note that constructing a PiecewisePolynomial requires at least two sample points, although in this case, the second sample point’s value is ignored, and only its break time is used.

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

class pydrake.trajectories.PiecewisePolynomial_[Expression]

Bases: pydrake.trajectories.PiecewiseTrajectory_[Expression]

A scalar multi-variate piecewise polynomial.

PiecewisePolynomial represents a list of contiguous segments in a scalar independent variable (typically corresponding to time) with Polynomials defined at each segment. We call the output from evaluating the PiecewisePolynomial at the scalar independent variable “the output”, and that output can be either a Eigen MatrixX<T> (if evaluated using value()) or a scalar (if evaluated using scalar_value()).

An example of a piecewise polynomial is a function of m segments in time, where a different polynomial is defined for each segment. For a specific example, consider the absolute value function over the interval [-1, 1]. We can define a PiecewisePolynomial over this interval using breaks at t = { -1.0, 0.0, 1.0 }, and “samples” of abs(t).

Click to expand C++ code...

// Construct the PiecewisePolynomial.
const std::vector<double> breaks = { -1.0, 0.0, 1.0 };
std::vector<Eigen::MatrixXd> samples(3);
for (int i = 0; i < static_cast<int>(breaks.size()); ++i) {
  samples[i].resize(1, 1);
  samples[i](0, 0) = std::abs(breaks[i]);
}
const auto pp =
     PiecewisePolynomial<double>::FirstOrderHold(breaks, samples);
const int row = 0, col = 0;

// Evaluate the PiecewisePolynomial at some values.
std::cout << pp.value(-.5)(row, col) << std::endl;    // Outputs 0.5.
std::cout << pp.value(0.0)(row, col) << std::endl;    // Outputs 0.0;

// Show how we can evaluate the first derivative (outputs -1.0).
std::cout << pp.derivative(1).value(-.5)(row, col) << std::endl;

A note on terminology. For piecewise-polynomial interpolation, we use breaks to indicate the scalar (e.g. times) which form the boundary of each segment. We use samples to indicate the function value at the breaks, e.g. p(breaks[i]) = samples[i]. The term knot should be reserved for the “(x,y)” coordinate, here knot[i] = (breaks[i], samples[i]), though it is used inconsistently in the interpolation literature (sometimes for breaks, sometimes for samples), so we try to mostly avoid it here.

PiecewisePolynomial objects can be added, subtracted, and multiplied. They cannot be divided because Polynomials are not closed under division.

Warning

PiecewisePolynomial silently clips input evaluations outside of the range defined by the breaks. So pp.value(-2.0, row, col) in the example above would evaluate to -1.0. See value().

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.PiecewisePolynomial_[Expression]) -> None

Constructs an empty piecewise polynomial.

__init__(self: pydrake.trajectories.PiecewisePolynomial_[Expression], arg0: numpy.ndarray[object[m, n], flags.f_contiguous]) -> None

Single segment, constant value constructor over the interval [-∞, ∞]. The constructed PiecewisePolynomial will return constant_value at every evaluated point (i.e., value(t) = constant_value ∀t ∈ [-∞, ∞]).

__init__(self: pydrake.trajectories.PiecewisePolynomial_[Expression], arg0: list[numpy.ndarray[object[m, n]]], arg1: list[pydrake.symbolic.Expression]) -> None

Constructs a PiecewisePolynomial using matrix-output Polynomials defined over each segment.

Precondition:: polynomials.size() == breaks.size() - 1

__init__(self: pydrake.trajectories.PiecewisePolynomial_[Expression], arg0: list[pydrake.polynomial.Polynomial_[Expression]], arg1: list[pydrake.symbolic.Expression]) -> None

Constructs a PiecewisePolynomial using scalar-output Polynomials defined over each segment.

Precondition:: polynomials.size() == breaks.size() - 1

AppendCubicHermiteSegment(self: pydrake.trajectories.PiecewisePolynomial_[Expression], time: pydrake.symbolic.Expression, sample: numpy.ndarray[object[m, n], flags.f_contiguous], sample_dot: numpy.ndarray[object[m, n], flags.f_contiguous]) → None

The CubicHermite spline construction has a nice property of being incremental (each segment can be solved independently). Given a new sample and it’s derivative, this method adds one segment to the end of this where the start sample and derivative are taken as the value and derivative at the final break of this.

Precondition:: this is not empty()
Precondition:: time > end_time()
Precondition:: sample and sample_dot must have size rows() x cols().

AppendFirstOrderSegment(self: pydrake.trajectories.PiecewisePolynomial_[Expression], time: pydrake.symbolic.Expression, sample: numpy.ndarray[object[m, n], flags.f_contiguous]) → None: Given a new sample, this method adds one segment to the end of this using a first-order hold, where the start sample is taken as the value at the final break of this.

Block(self: pydrake.trajectories.PiecewisePolynomial_[Expression], start_row: int, start_col: int, block_rows: int, block_cols: int) → pydrake.trajectories.PiecewisePolynomial_[Expression]

Extracts a trajectory representing a block of size (block_rows, block_cols) starting at (start_row, start_col) from the PiecewisePolynomial.

Returns: a PiecewisePolynomial such that ret.value(t) = this.value(t).block(i,j,p,q);

Clone(self: pydrake.trajectories.PiecewisePolynomial_[Expression]) → pydrake.trajectories.Trajectory_[Expression]

ConcatenateInTime(self: pydrake.trajectories.PiecewisePolynomial_[Expression], other: pydrake.trajectories.PiecewisePolynomial_[Expression]) → None

Concatenates other to the end of this.

Warning

The resulting PiecewisePolynomial will only be continuous to the degree that the first Polynomial of other is continuous with the last Polynomial of this. See warning about evaluating discontinuous derivatives at breaks in derivative().

Parameter other:: PiecewisePolynomial instance to concatenate.

Raises

RuntimeError if trajectories' dimensions do not match each other –
(either rows() or cols() does not match between this and –
other`) –
RuntimeError if this->end_time() and other->start_time() –
are not within PiecewiseTrajectory<T>::kEpsilonTime from each –
other. –

static CubicHermite(*args, **kwargs)

Overloaded function.

CubicHermite(breaks: list[pydrake.symbolic.Expression], samples: list[list[pydrake.symbolic.Expression]], samples_dot: list[list[pydrake.symbolic.Expression]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Version of CubicHermite(breaks, samples, samples_dot) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Corresponding columns of samples and samples_dot are used as the sample point and independent variable derivative, respectively.

Precondition:: samples.cols() == samples_dot.cols() == breaks.size().

CubicHermite(breaks: list[pydrake.symbolic.Expression], samples: list[numpy.ndarray[object[m, n]]], samples_dot: list[numpy.ndarray[object[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Constructs a third order PiecewisePolynomial using matrix samples and derivatives of samples (samples_dot); each matrix element of samples_dot represents the derivative with respect to the independent variable (e.g., the time derivative) of the corresponding entry in samples. Each segment is fully specified by samples and sample_dot at both ends. Second derivatives are not continuous.

static CubicShapePreserving(*args, **kwargs)

Overloaded function.

CubicShapePreserving(breaks: list[pydrake.symbolic.Expression], samples: list[list[pydrake.symbolic.Expression]], zero_end_point_derivatives: bool = False) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Version of CubicShapePreserving(breaks, samples, zero_end_point_derivatives) that uses vector samples and Eigen VectorXd and MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size().

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

CubicShapePreserving(breaks: list[pydrake.symbolic.Expression], samples: list[numpy.ndarray[object[m, n]]], zero_end_point_derivatives: bool = False) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Constructs a third order PiecewisePolynomial using vector samples, where each column of samples represents a sample point. First derivatives are chosen to be “shape preserving”, i.e. if samples is monotonic within some interval, the interpolated data will also be monotonic. The second derivative is not guaranteed to be smooth across the entire spline.

MATLAB calls this method “pchip” (short for “Piecewise Cubic Hermite Interpolating Polynomial”), and provides a nice description in their documentation. http://home.uchicago.edu/~sctchoi/courses/cs138/interp.pdf is also a good reference.

If zero_end_point_derivatives is False, the first and last first derivative is chosen using a non-centered, shape-preserving three-point formulae. See equation (2.10) in the following reference for more details. http://www.mi.sanu.ac.rs/~gvm/radovi/mon.pdf If zero_end_point_derivatives is True, they are set to zeros.

If zero_end_point_derivatives is False, breaks and samples must have at least 3 elements for the algorithm to determine the first derivatives.

If zero_end_point_derivatives is True, breaks and samples may have 2 or more elements. For the 2 elements case, the result is equivalent to computing a cubic polynomial whose values are given by samples, and derivatives set to zero.

Raises

RuntimeError if –
- breaks has length smaller than 3 and
zero_end_point_derivatives` is False, - breaks has lengt –
smaller than 2 and zero_end_point_derivatives is true. –
RuntimeError under the conditions specified under –
coefficient_construction_methods. –

static CubicWithContinuousSecondDerivatives(*args, **kwargs)

Overloaded function.

CubicWithContinuousSecondDerivatives(breaks: list[pydrake.symbolic.Expression], samples: list[list[pydrake.symbolic.Expression]], sample_dot_at_start: numpy.ndarray[object[m, n]], sample_dot_at_end: numpy.ndarray[object[m, n]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Version of CubicWithContinuousSecondDerivatives() that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size().

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

CubicWithContinuousSecondDerivatives(breaks: list[pydrake.symbolic.Expression], samples: list[numpy.ndarray[object[m, n]]], sample_dot_at_start: numpy.ndarray[object[m, n]], sample_dot_at_end: numpy.ndarray[object[m, n]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Constructs a third order PiecewisePolynomial using matrix samples. The PiecewisePolynomial is constructed such that the interior segments have the same value, first and second derivatives at breaks. sample_dot_at_start and sample_dot_at_end are used for the first and last first derivatives.

Raises

RuntimeError if sample_dot_at_start or sample_dot_at_end –
and samples have inconsistent dimensions. –
RuntimeError under the conditions specified under –
coefficient_construction_methods. –

CubicWithContinuousSecondDerivatives(breaks: list[pydrake.symbolic.Expression], samples: list[list[pydrake.symbolic.Expression]], periodic_end_condition: bool = False) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Version of CubicWithContinuousSecondDerivatives(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size().

CubicWithContinuousSecondDerivatives(breaks: list[pydrake.symbolic.Expression], samples: list[numpy.ndarray[object[m, n]]], periodic_end: bool) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Constructs a third order PiecewisePolynomial using matrix samples. The PiecewisePolynomial is constructed such that the interior segments have the same value, first and second derivatives at breaks. If periodic_end_condition is False (default), then the “Not-a-sample” end condition is used here, which means the third derivatives are continuous for the first two and last two segments. If periodic_end_condition is True, then the first and second derivatives between the end of the last segment and the beginning of the first segment will be continuous. Note that the periodic end condition does not require the first and last sample to be collocated, nor does it add an additional sample to connect the first and last segments. Only first and second derivative continuity is enforced. See https://en.wikipedia.org/wiki/Spline_interpolation and https://web.archive.org/web/20140125011904/https://www.math.uh.edu/~jingqiu/math4364/spline.pdf for more about cubic splines and their end conditions. The MATLAB docs for methods “spline” and “csape” are also good references.

Precondition:: breaks and samples must have at least 3 elements. If periodic_end_condition is True, then for two samples, it would produce a straight line (use FirstOrderHold for this instead), and if periodic_end_condition is False the problem is ill-defined.

derivative(self: pydrake.trajectories.PiecewisePolynomial_[Expression], derivative_order: int = 1) → pydrake.trajectories.PiecewisePolynomial_[Expression]

Returns a PiecewisePolynomial where each segment is the specified derivative of the corresponding segment in this. Any rules or limitations of Polynomial::derivative() also apply to this function.

Derivatives evaluated at non-differentiable points return the value at the left hand side of the interval.

Parameter derivative_order:: The order of the derivative, namely, if derivative_order = n, the n’th derivative of the polynomial will be returned.

Warning

In the event of discontinuous derivatives evaluated at breaks, it is not defined which polynomial (i.e., to the left or right of the break) will be the one that is evaluated at the break.

static FirstOrderHold(*args, **kwargs)

Overloaded function.

FirstOrderHold(breaks: list[pydrake.symbolic.Expression], samples: list[list[pydrake.symbolic.Expression]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Version of FirstOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size()

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

FirstOrderHold(breaks: list[pydrake.symbolic.Expression], samples: list[numpy.ndarray[object[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Constructs a piecewise linear PiecewisePolynomial using matrix samples.

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

getPolynomial(self: pydrake.trajectories.PiecewisePolynomial_[Expression], segment_index: int, row: int = 0, col: int = 0) → pydrake.polynomial.Polynomial_[Expression]: Gets the Polynomial with the given matrix row and column index that corresponds to the given segment index. Equivalent to getPolynomialMatrix(segment_index)(row, col).

Note

Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

getPolynomialMatrix(self: pydrake.trajectories.PiecewisePolynomial_[Expression], segment_index: int) → numpy.ndarray[object[m, n]]: Gets the matrix of Polynomials corresponding to the given segment index.

Warning

segment_index is not checked for validity.

getSegmentPolynomialDegree(self: pydrake.trajectories.PiecewisePolynomial_[Expression], segment_index: int, row: int = 0, col: int = 0) → int: Gets the degree of the Polynomial with the given matrix row and column index that corresponds to the given segment index. Equivalent to getPolynomial(segment_index, row, col).GetDegree().

isApprox(self: pydrake.trajectories.PiecewisePolynomial_[Expression], other: pydrake.trajectories.PiecewisePolynomial_[Expression], tol: float, tol_type: pydrake.common.ToleranceType = <ToleranceType.kRelative: 1>) → bool: Checks whether a PiecewisePolynomial is approximately equal to this one by calling Polynomial<T>::CoefficientsAlmostEqual() on every element of every segment.

See also

Polynomial<T>::CoefficientsAlmostEqual().

static LagrangeInterpolatingPolynomial(*args, **kwargs)

Overloaded function.

LagrangeInterpolatingPolynomial(times: list[pydrake.symbolic.Expression], samples: list[list[pydrake.symbolic.Expression]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Version of LagrangeInterpolatingPolynomial(times, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == times.size().

LagrangeInterpolatingPolynomial(times: list[pydrake.symbolic.Expression], samples: list[numpy.ndarray[object[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Constructs a polynomial with a single segment of the lowest possible degree that passes through all of the sample points. See “polynomial interpolation” and/or “Lagrange polynomial” on Wikipedia for more information.

Precondition:: times must be monotonically increasing.
Precondition:: samples.size() == times.size().

RemoveFinalSegment(self: pydrake.trajectories.PiecewisePolynomial_[Expression]) → None

Removes the final segment from the trajectory, reducing the number of segments by 1.

Precondition:: this is not empty()

Reshape(self: pydrake.trajectories.PiecewisePolynomial_[Expression], rows: int, cols: int) → None

Reshapes the dimensions of the Eigen::MatrixX<T> returned by value(), EvalDerivative(), etc.

Precondition:: rows x cols must equal this.rows() * this.cols().

See also

Eigen::PlainObjectBase::resize().

ReverseTime(self: pydrake.trajectories.PiecewisePolynomial_[Expression]) → None: Modifies the trajectory so that pp_after(t) = pp_before(-t).

Note

The new trajectory will evaluate differently at precisely the break points if the original trajectory was discontinuous at the break points. This is because the segments are defined on the half-open intervals [breaks(i), breaks(i+1)), and the order of the breaks have been reversed.

ScaleTime(self: pydrake.trajectories.PiecewisePolynomial_[Expression], scale: pydrake.symbolic.Expression) → None

Scales the time of the trajectory by non-negative scale (use ReverseTime() if you want to also negate time). The resulting polynomial evaluates to pp_after(t) = pp_before(t/scale).

As an example, `scale`=2 will result in a trajectory that is twice as long (start_time() and end_time() have both doubled).

setPolynomialMatrixBlock(self: pydrake.trajectories.PiecewisePolynomial_[Expression], replacement: numpy.ndarray[object[m, n]], segment_index: int, row_start: int = 0, col_start: int = 0) → None: Replaces the specified block of the PolynomialMatrix at the given segment index.

Note

Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

Warning

This code relies upon Eigen to verify that the replacement block is not too large.

shiftRight(self: pydrake.trajectories.PiecewisePolynomial_[Expression], offset: pydrake.symbolic.Expression) → None

Adds offset to all of the breaks. offset need not be a non-negative number. The resulting polynomial will evaluate to pp_after(t) = pp_before(t-offset).

As an example, `offset`=2 will result in the start_time() and end_time() being 2 seconds later.

slice(self: pydrake.trajectories.PiecewisePolynomial_[Expression], start_segment_index: int, num_segments: int) → pydrake.trajectories.PiecewisePolynomial_[Expression]: Returns the PiecewisePolynomial comprising the num_segments segments starting at the specified start_segment_index.

Note

Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

Transpose(self: pydrake.trajectories.PiecewisePolynomial_[Expression]) → pydrake.trajectories.PiecewisePolynomial_[Expression]: Constructs a new PiecewisePolynomial for which value(t) == this.value(t).transpose().

static ZeroOrderHold(*args, **kwargs)

Overloaded function.

ZeroOrderHold(breaks: list[pydrake.symbolic.Expression], samples: list[list[pydrake.symbolic.Expression]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Version of ZeroOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd/MatrixX<T> arguments. Each column of samples represents a sample point.

Precondition:: samples.cols() == breaks.size()

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

ZeroOrderHold(breaks: list[pydrake.symbolic.Expression], samples: list[numpy.ndarray[object[m, n]]]) -> pydrake.trajectories.PiecewisePolynomial_[Expression]

Constructs a piecewise constant PiecewisePolynomial using matrix samples. Note that constructing a PiecewisePolynomial requires at least two sample points, although in this case, the second sample point’s value is ignored, and only its break time is used.

Raises

RuntimeError under the conditions specified under –
coefficient_construction_methods. –

class pydrake.trajectories.PiecewisePose

Bases: pydrake.trajectories.PiecewiseTrajectory

A wrapper class that represents a pose trajectory, whose rotation part is a PiecewiseQuaternionSlerp and the translation part is a PiecewisePolynomial.

Note

This class is templated; see PiecewisePose_ for the list of instantiations.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.PiecewisePose) -> None

Constructs an empty piecewise pose trajectory.

__init__(self: pydrake.trajectories.PiecewisePose, position_trajectory: pydrake.trajectories.PiecewisePolynomial, orientation_trajectory: pydrake.trajectories.PiecewiseQuaternionSlerp) -> None

Constructor.

Parameter pos_traj:: Position trajectory.
Parameter rot_traj:: Orientation trajectory.

get_orientation_trajectory(self: pydrake.trajectories.PiecewisePose) → pydrake.trajectories.PiecewiseQuaternionSlerp: Returns the orientation trajectory.

get_position_trajectory(self: pydrake.trajectories.PiecewisePose) → pydrake.trajectories.PiecewisePolynomial: Returns the position trajectory.

GetAcceleration(self: pydrake.trajectories.PiecewisePose, time: float) → numpy.ndarray[numpy.float64[6, 1]]: Returns the interpolated acceleration at time or zero if time is before this trajectory’s start time or after its end time.

GetPose(self: pydrake.trajectories.PiecewisePose, time: float) → pydrake.math.RigidTransform: Returns the interpolated pose at time.

GetVelocity(self: pydrake.trajectories.PiecewisePose, time: float) → numpy.ndarray[numpy.float64[6, 1]]: Returns the interpolated velocity at time or zero if time is before this trajectory’s start time or after its end time.

IsApprox(self: pydrake.trajectories.PiecewisePose, other: pydrake.trajectories.PiecewisePose, tol: float) → bool: Returns true if the position and orientation trajectories are both within tol from the other’s.

static MakeCubicLinearWithEndLinearVelocity(times: list[float], poses: list[pydrake.math.RigidTransform], start_vel: numpy.ndarray[numpy.float64[3, 1]] = array([0., 0., 0.]), end_vel: numpy.ndarray[numpy.float64[3, 1]] = array([0., 0., 0.])) → pydrake.trajectories.PiecewisePose

Constructs a PiecewisePose from given times and poses. A cubic polynomial with given end velocities is used to construct the position part. The rotational part is represented by a piecewise quaterion trajectory. There must be at least two elements in times and poses.

Parameter times:: Breaks used to build the splines.
Parameter poses:: Knots used to build the splines.
Parameter start_vel:: Start linear velocity.
Parameter end_vel:: End linear velocity.

static MakeLinear(times: list[float], poses: list[pydrake.math.RigidTransform]) → pydrake.trajectories.PiecewisePose

Constructs a PiecewisePose from given times and poses. The positions trajectory is constructed as a first-order hold. The orientation is constructed using the quaternion slerp. There must be at least two elements in times and poses.

Parameter times:: Breaks used to build the splines.
Parameter poses:: Knots used to build the splines.

template pydrake.trajectories.PiecewisePose_: Instantiations: PiecewisePose_[float], PiecewisePose_[AutoDiffXd], PiecewisePose_[Expression]

class pydrake.trajectories.PiecewisePose_[AutoDiffXd]

Bases: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd]

A wrapper class that represents a pose trajectory, whose rotation part is a PiecewiseQuaternionSlerp and the translation part is a PiecewisePolynomial.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.PiecewisePose_[AutoDiffXd]) -> None

Constructs an empty piecewise pose trajectory.

__init__(self: pydrake.trajectories.PiecewisePose_[AutoDiffXd], position_trajectory: pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd], orientation_trajectory: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd]) -> None

Constructor.

Parameter pos_traj:: Position trajectory.
Parameter rot_traj:: Orientation trajectory.

get_orientation_trajectory(self: pydrake.trajectories.PiecewisePose_[AutoDiffXd]) → pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd]: Returns the orientation trajectory.

get_position_trajectory(self: pydrake.trajectories.PiecewisePose_[AutoDiffXd]) → pydrake.trajectories.PiecewisePolynomial_[AutoDiffXd]: Returns the position trajectory.

GetAcceleration(self: pydrake.trajectories.PiecewisePose_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd) → numpy.ndarray[object[6, 1]]: Returns the interpolated acceleration at time or zero if time is before this trajectory’s start time or after its end time.

GetPose(self: pydrake.trajectories.PiecewisePose_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd) → pydrake.math.RigidTransform_[AutoDiffXd]: Returns the interpolated pose at time.

GetVelocity(self: pydrake.trajectories.PiecewisePose_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd) → numpy.ndarray[object[6, 1]]: Returns the interpolated velocity at time or zero if time is before this trajectory’s start time or after its end time.

IsApprox(self: pydrake.trajectories.PiecewisePose_[AutoDiffXd], other: pydrake.trajectories.PiecewisePose_[AutoDiffXd], tol: float) → bool: Returns true if the position and orientation trajectories are both within tol from the other’s.

static MakeCubicLinearWithEndLinearVelocity(times: list[pydrake.autodiffutils.AutoDiffXd], poses: list[pydrake.math.RigidTransform_[AutoDiffXd]], start_vel: numpy.ndarray[object[3, 1]] = array([<AutoDiffXd 0.0 nderiv=0>, <AutoDiffXd 0.0 nderiv=0>, <AutoDiffXd 0.0 nderiv=0>], dtype=object), end_vel: numpy.ndarray[object[3, 1]] = array([<AutoDiffXd 0.0 nderiv=0>, <AutoDiffXd 0.0 nderiv=0>, <AutoDiffXd 0.0 nderiv=0>], dtype=object)) → pydrake.trajectories.PiecewisePose_[AutoDiffXd]

Constructs a PiecewisePose from given times and poses. A cubic polynomial with given end velocities is used to construct the position part. The rotational part is represented by a piecewise quaterion trajectory. There must be at least two elements in times and poses.

Parameter times:: Breaks used to build the splines.
Parameter poses:: Knots used to build the splines.
Parameter start_vel:: Start linear velocity.
Parameter end_vel:: End linear velocity.

static MakeLinear(times: list[pydrake.autodiffutils.AutoDiffXd], poses: list[pydrake.math.RigidTransform_[AutoDiffXd]]) → pydrake.trajectories.PiecewisePose_[AutoDiffXd]

Constructs a PiecewisePose from given times and poses. The positions trajectory is constructed as a first-order hold. The orientation is constructed using the quaternion slerp. There must be at least two elements in times and poses.

Parameter times:: Breaks used to build the splines.
Parameter poses:: Knots used to build the splines.

class pydrake.trajectories.PiecewisePose_[Expression]

Bases: pydrake.trajectories.PiecewiseTrajectory_[Expression]

A wrapper class that represents a pose trajectory, whose rotation part is a PiecewiseQuaternionSlerp and the translation part is a PiecewisePolynomial.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.PiecewisePose_[Expression]) -> None

Constructs an empty piecewise pose trajectory.

__init__(self: pydrake.trajectories.PiecewisePose_[Expression], position_trajectory: pydrake.trajectories.PiecewisePolynomial_[Expression], orientation_trajectory: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression]) -> None

Constructor.

Parameter pos_traj:: Position trajectory.
Parameter rot_traj:: Orientation trajectory.

get_orientation_trajectory(self: pydrake.trajectories.PiecewisePose_[Expression]) → pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression]: Returns the orientation trajectory.

get_position_trajectory(self: pydrake.trajectories.PiecewisePose_[Expression]) → pydrake.trajectories.PiecewisePolynomial_[Expression]: Returns the position trajectory.

GetAcceleration(self: pydrake.trajectories.PiecewisePose_[Expression], time: pydrake.symbolic.Expression) → numpy.ndarray[object[6, 1]]: Returns the interpolated acceleration at time or zero if time is before this trajectory’s start time or after its end time.

GetPose(self: pydrake.trajectories.PiecewisePose_[Expression], time: pydrake.symbolic.Expression) → pydrake.math.RigidTransform_[Expression]: Returns the interpolated pose at time.

GetVelocity(self: pydrake.trajectories.PiecewisePose_[Expression], time: pydrake.symbolic.Expression) → numpy.ndarray[object[6, 1]]: Returns the interpolated velocity at time or zero if time is before this trajectory’s start time or after its end time.

IsApprox(self: pydrake.trajectories.PiecewisePose_[Expression], other: pydrake.trajectories.PiecewisePose_[Expression], tol: float) → bool: Returns true if the position and orientation trajectories are both within tol from the other’s.

static MakeCubicLinearWithEndLinearVelocity(times: list[pydrake.symbolic.Expression], poses: list[pydrake.math.RigidTransform_[Expression]], start_vel: numpy.ndarray[object[3, 1]] = array([<Expression "0">, <Expression "0">, <Expression "0">], dtype=object), end_vel: numpy.ndarray[object[3, 1]] = array([<Expression "0">, <Expression "0">, <Expression "0">], dtype=object)) → pydrake.trajectories.PiecewisePose_[Expression]

Constructs a PiecewisePose from given times and poses. A cubic polynomial with given end velocities is used to construct the position part. The rotational part is represented by a piecewise quaterion trajectory. There must be at least two elements in times and poses.

Parameter times:: Breaks used to build the splines.
Parameter poses:: Knots used to build the splines.
Parameter start_vel:: Start linear velocity.
Parameter end_vel:: End linear velocity.

static MakeLinear(times: list[pydrake.symbolic.Expression], poses: list[pydrake.math.RigidTransform_[Expression]]) → pydrake.trajectories.PiecewisePose_[Expression]

Constructs a PiecewisePose from given times and poses. The positions trajectory is constructed as a first-order hold. The orientation is constructed using the quaternion slerp. There must be at least two elements in times and poses.

Parameter times:: Breaks used to build the splines.
Parameter poses:: Knots used to build the splines.

class pydrake.trajectories.PiecewiseQuaternionSlerp

Bases: pydrake.trajectories.PiecewiseTrajectory

A class representing a trajectory for quaternions that are interpolated using piecewise slerp (spherical linear interpolation). All the orientation samples are expected to be with respect to the same parent reference frame, i.e. q_i represents the rotation R_PBi for the orientation of frame B at the ith sample in a fixed parent frame P. The world frame is a common choice for the parent frame. The angular velocity and acceleration are also relative to the parent frame and expressed in the parent frame. Since there is a sign ambiguity when using quaternions to represent orientation, namely q and -q represent the same orientation, the internal quaternion representations ensure that q_n.dot(q_{n+1}) >= 0. Another intuitive way to think about this is that consecutive quaternions have the shortest geodesic distance on the unit sphere. Note that the quarternion value is in w, x, y, z order when represented as a Vector4.

Note

This class is templated; see PiecewiseQuaternionSlerp_ for the list of instantiations.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp) -> None

Builds an empty PiecewiseQuaternionSlerp.

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp, breaks: list[float], quaternions: list[pydrake.common.eigen_geometry.Quaternion]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and quaternions have different length, or –
breaks have length < 2. –

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp, breaks: list[float], rotation_matrices: list[numpy.ndarray[numpy.float64[3, 3]]]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and rot_matrices have different length, or –
breaks have length < 2. –

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp, breaks: list[float], rotation_matrices: list[pydrake.math.RotationMatrix]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and rot_matrices have different length, or –
breaks have length < 2. –

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp, breaks: list[float], angle_axes: list[pydrake.common.eigen_geometry.AngleAxis]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and ang_axes have different length, or –
breaks have length < 2. –

angular_acceleration(self: pydrake.trajectories.PiecewiseQuaternionSlerp, time: float) → numpy.ndarray[numpy.float64[3, 1]]

Interpolates angular acceleration.

Parameter time:: Time for interpolation.

Returns: The interpolated angular acceleration at time, which is always zero for slerp.

angular_velocity(self: pydrake.trajectories.PiecewiseQuaternionSlerp, time: float) → numpy.ndarray[numpy.float64[3, 1]]

Interpolates angular velocity.

Parameter time:: Time for interpolation.

Returns: The interpolated angular velocity at time, which is constant per segment.

Append(*args, **kwargs)

Overloaded function.

Append(self: pydrake.trajectories.PiecewiseQuaternionSlerp, time: float, quaternion: pydrake.common.eigen_geometry.Quaternion) -> None

Given a new Quaternion, this method adds one segment to the end of this.

Append(self: pydrake.trajectories.PiecewiseQuaternionSlerp, time: float, rotation_matrix: pydrake.math.RotationMatrix) -> None

Given a new RotationMatrix, this method adds one segment to the end of this.

Append(self: pydrake.trajectories.PiecewiseQuaternionSlerp, time: float, angle_axis: pydrake.common.eigen_geometry.AngleAxis) -> None

Given a new AngleAxis, this method adds one segment to the end of this.

orientation(self: pydrake.trajectories.PiecewiseQuaternionSlerp, time: float) → pydrake.common.eigen_geometry.Quaternion

Interpolates orientation.

Parameter time:: Time for interpolation.

Returns: The interpolated quaternion at time.

template pydrake.trajectories.PiecewiseQuaternionSlerp_: Instantiations: PiecewiseQuaternionSlerp_[float], PiecewiseQuaternionSlerp_[AutoDiffXd], PiecewiseQuaternionSlerp_[Expression]

class pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd]

Bases: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd]

A class representing a trajectory for quaternions that are interpolated using piecewise slerp (spherical linear interpolation). All the orientation samples are expected to be with respect to the same parent reference frame, i.e. q_i represents the rotation R_PBi for the orientation of frame B at the ith sample in a fixed parent frame P. The world frame is a common choice for the parent frame. The angular velocity and acceleration are also relative to the parent frame and expressed in the parent frame. Since there is a sign ambiguity when using quaternions to represent orientation, namely q and -q represent the same orientation, the internal quaternion representations ensure that q_n.dot(q_{n+1}) >= 0. Another intuitive way to think about this is that consecutive quaternions have the shortest geodesic distance on the unit sphere. Note that the quarternion value is in w, x, y, z order when represented as a Vector4.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd]) -> None

Builds an empty PiecewiseQuaternionSlerp.

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], breaks: list[pydrake.autodiffutils.AutoDiffXd], quaternions: list[pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and quaternions have different length, or –
breaks have length < 2. –

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], breaks: list[pydrake.autodiffutils.AutoDiffXd], rotation_matrices: list[numpy.ndarray[object[3, 3]]]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and rot_matrices have different length, or –
breaks have length < 2. –

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], breaks: list[pydrake.autodiffutils.AutoDiffXd], rotation_matrices: list[pydrake.math.RotationMatrix_[AutoDiffXd]]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and rot_matrices have different length, or –
breaks have length < 2. –

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], breaks: list[pydrake.autodiffutils.AutoDiffXd], angle_axes: list[pydrake.common.eigen_geometry.AngleAxis_[AutoDiffXd]]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and ang_axes have different length, or –
breaks have length < 2. –

angular_acceleration(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd) → numpy.ndarray[object[3, 1]]

Interpolates angular acceleration.

Parameter time:: Time for interpolation.

Returns: The interpolated angular acceleration at time, which is always zero for slerp.

angular_velocity(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd) → numpy.ndarray[object[3, 1]]

Interpolates angular velocity.

Parameter time:: Time for interpolation.

Returns: The interpolated angular velocity at time, which is constant per segment.

Append(*args, **kwargs)

Overloaded function.

Append(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd, quaternion: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]) -> None

Given a new Quaternion, this method adds one segment to the end of this.

Append(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd, rotation_matrix: pydrake.math.RotationMatrix_[AutoDiffXd]) -> None

Given a new RotationMatrix, this method adds one segment to the end of this.

Append(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd, angle_axis: pydrake.common.eigen_geometry.AngleAxis_[AutoDiffXd]) -> None

Given a new AngleAxis, this method adds one segment to the end of this.

orientation(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[AutoDiffXd], time: pydrake.autodiffutils.AutoDiffXd) → pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]

Interpolates orientation.

Parameter time:: Time for interpolation.

Returns: The interpolated quaternion at time.

class pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression]

Bases: pydrake.trajectories.PiecewiseTrajectory_[Expression]

A class representing a trajectory for quaternions that are interpolated using piecewise slerp (spherical linear interpolation). All the orientation samples are expected to be with respect to the same parent reference frame, i.e. q_i represents the rotation R_PBi for the orientation of frame B at the ith sample in a fixed parent frame P. The world frame is a common choice for the parent frame. The angular velocity and acceleration are also relative to the parent frame and expressed in the parent frame. Since there is a sign ambiguity when using quaternions to represent orientation, namely q and -q represent the same orientation, the internal quaternion representations ensure that q_n.dot(q_{n+1}) >= 0. Another intuitive way to think about this is that consecutive quaternions have the shortest geodesic distance on the unit sphere. Note that the quarternion value is in w, x, y, z order when represented as a Vector4.

__init__(*args, **kwargs)

Overloaded function.

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression]) -> None

Builds an empty PiecewiseQuaternionSlerp.

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], breaks: list[pydrake.symbolic.Expression], quaternions: list[pydrake.common.eigen_geometry.Quaternion_[Expression]]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and quaternions have different length, or –
breaks have length < 2. –

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], breaks: list[pydrake.symbolic.Expression], rotation_matrices: list[numpy.ndarray[object[3, 3]]]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and rot_matrices have different length, or –
breaks have length < 2. –

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], breaks: list[pydrake.symbolic.Expression], rotation_matrices: list[pydrake.math.RotationMatrix_[Expression]]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and rot_matrices have different length, or –
breaks have length < 2. –

__init__(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], breaks: list[pydrake.symbolic.Expression], angle_axes: list[pydrake.common.eigen_geometry.AngleAxis_[Expression]]) -> None

Builds a PiecewiseQuaternionSlerp.

Raises

RuntimeError if breaks and ang_axes have different length, or –
breaks have length < 2. –

angular_acceleration(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], time: pydrake.symbolic.Expression) → numpy.ndarray[object[3, 1]]

Interpolates angular acceleration.

Parameter time:: Time for interpolation.

Returns: The interpolated angular acceleration at time, which is always zero for slerp.

angular_velocity(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], time: pydrake.symbolic.Expression) → numpy.ndarray[object[3, 1]]

Interpolates angular velocity.

Parameter time:: Time for interpolation.

Returns: The interpolated angular velocity at time, which is constant per segment.

Append(*args, **kwargs)

Overloaded function.

Append(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], time: pydrake.symbolic.Expression, quaternion: pydrake.common.eigen_geometry.Quaternion_[Expression]) -> None

Given a new Quaternion, this method adds one segment to the end of this.

Append(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], time: pydrake.symbolic.Expression, rotation_matrix: pydrake.math.RotationMatrix_[Expression]) -> None

Given a new RotationMatrix, this method adds one segment to the end of this.

Append(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], time: pydrake.symbolic.Expression, angle_axis: pydrake.common.eigen_geometry.AngleAxis_[Expression]) -> None

Given a new AngleAxis, this method adds one segment to the end of this.

orientation(self: pydrake.trajectories.PiecewiseQuaternionSlerp_[Expression], time: pydrake.symbolic.Expression) → pydrake.common.eigen_geometry.Quaternion_[Expression]

Interpolates orientation.

Parameter time:: Time for interpolation.

Returns: The interpolated quaternion at time.

class pydrake.trajectories.PiecewiseTrajectory

Bases: pydrake.trajectories.Trajectory

Abstract class that implements the basic logic of maintaining consequent segments of time (delimited by breaks) to implement a trajectory that is represented by simpler logic in each segment or “piece”.

Note

This class is templated; see PiecewiseTrajectory_ for the list of instantiations.

__init__(*args, **kwargs)

duration(self: pydrake.trajectories.PiecewiseTrajectory, segment_index: int) → float

end_time(*args, **kwargs)

Overloaded function.

end_time(self: pydrake.trajectories.PiecewiseTrajectory, segment_index: int) -> float
end_time(self: pydrake.trajectories.PiecewiseTrajectory) -> float

get_number_of_segments(self: pydrake.trajectories.PiecewiseTrajectory) → int

get_segment_index(self: pydrake.trajectories.PiecewiseTrajectory, t: float) → int

get_segment_times(self: pydrake.trajectories.PiecewiseTrajectory) → list[float]

is_time_in_range(self: pydrake.trajectories.PiecewiseTrajectory, t: float) → bool: Returns true iff t >= getStartTime() && t <= getEndTime().

start_time(*args, **kwargs)

Overloaded function.

start_time(self: pydrake.trajectories.PiecewiseTrajectory, segment_index: int) -> float
start_time(self: pydrake.trajectories.PiecewiseTrajectory) -> float

template pydrake.trajectories.PiecewiseTrajectory_: Instantiations: PiecewiseTrajectory_[float], PiecewiseTrajectory_[AutoDiffXd], PiecewiseTrajectory_[Expression]

class pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd]

Bases: pydrake.trajectories.Trajectory_[AutoDiffXd]

Abstract class that implements the basic logic of maintaining consequent segments of time (delimited by breaks) to implement a trajectory that is represented by simpler logic in each segment or “piece”.

__init__(*args, **kwargs)

duration(self: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd], segment_index: int) → pydrake.autodiffutils.AutoDiffXd

end_time(*args, **kwargs)

Overloaded function.

end_time(self: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd], segment_index: int) -> pydrake.autodiffutils.AutoDiffXd
end_time(self: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd]) -> pydrake.autodiffutils.AutoDiffXd

get_number_of_segments(self: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd]) → int

get_segment_index(self: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd], t: pydrake.autodiffutils.AutoDiffXd) → int

get_segment_times(self: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd]) → list[pydrake.autodiffutils.AutoDiffXd]

is_time_in_range(self: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd], t: pydrake.autodiffutils.AutoDiffXd) → bool: Returns true iff t >= getStartTime() && t <= getEndTime().

start_time(*args, **kwargs)

Overloaded function.

start_time(self: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd], segment_index: int) -> pydrake.autodiffutils.AutoDiffXd
start_time(self: pydrake.trajectories.PiecewiseTrajectory_[AutoDiffXd]) -> pydrake.autodiffutils.AutoDiffXd

class pydrake.trajectories.PiecewiseTrajectory_[Expression]

Bases: pydrake.trajectories.Trajectory_[Expression]

Abstract class that implements the basic logic of maintaining consequent segments of time (delimited by breaks) to implement a trajectory that is represented by simpler logic in each segment or “piece”.

__init__(*args, **kwargs)

duration(self: pydrake.trajectories.PiecewiseTrajectory_[Expression], segment_index: int) → pydrake.symbolic.Expression

end_time(*args, **kwargs)

Overloaded function.

end_time(self: pydrake.trajectories.PiecewiseTrajectory_[Expression], segment_index: int) -> pydrake.symbolic.Expression
end_time(self: pydrake.trajectories.PiecewiseTrajectory_[Expression]) -> pydrake.symbolic.Expression

get_number_of_segments(self: pydrake.trajectories.PiecewiseTrajectory_[Expression]) → int

get_segment_index(self: pydrake.trajectories.PiecewiseTrajectory_[Expression], t: pydrake.symbolic.Expression) → int

get_segment_times(self: pydrake.trajectories.PiecewiseTrajectory_[Expression]) → list[pydrake.symbolic.Expression]

is_time_in_range(self: pydrake.trajectories.PiecewiseTrajectory_[Expression], t: pydrake.symbolic.Expression) → pydrake.symbolic.Formula: Returns true iff t >= getStartTime() && t <= getEndTime().

start_time(*args, **kwargs)

Overloaded function.

start_time(self: pydrake.trajectories.PiecewiseTrajectory_[Expression], segment_index: int) -> pydrake.symbolic.Expression
start_time(self: pydrake.trajectories.PiecewiseTrajectory_[Expression]) -> pydrake.symbolic.Expression

class pydrake.trajectories.StackedTrajectory

Bases: pydrake.trajectories.Trajectory

A StackedTrajectory stacks the values from one or more underlying Trajectory objects into a single Trajectory, without changing the %start_time() or %end_time().

For sequencing trajectories in time instead, see CompositeTrajectory.

All of the underlying Trajectory objects must have the same %start_time() and %end_time().

When constructed with rowwise set to true, all of the underlying Trajectory objects must have the same number of %cols() and the value() matrix will be the vstack of the the trajectories in the order they were added.

When constructed with rowwise set to false, all of the underlying Trajectory objects must have the same number of %rows() and the value() matrix will be the hstack of the the trajectories in the order they were added.

Note

This class is templated; see StackedTrajectory_ for the list of instantiations.

__init__(self: pydrake.trajectories.StackedTrajectory, rowwise: bool = True) → None

Creates an empty trajectory.

Parameter rowwise:: governs the stacking order

Append(self: pydrake.trajectories.StackedTrajectory, arg0: pydrake.trajectories.Trajectory) → None

Stacks another sub-Trajectory onto this. Refer to the class overview documentation for details.

Raises: RuntimeError if the matrix dimension is incompatible. –

Clone(self: pydrake.trajectories.StackedTrajectory) → pydrake.trajectories.Trajectory

template pydrake.trajectories.StackedTrajectory_: Instantiations: StackedTrajectory_[float], StackedTrajectory_[AutoDiffXd], StackedTrajectory_[Expression]

class pydrake.trajectories.StackedTrajectory_[AutoDiffXd]

Bases: pydrake.trajectories.Trajectory_[AutoDiffXd]

A StackedTrajectory stacks the values from one or more underlying Trajectory objects into a single Trajectory, without changing the %start_time() or %end_time().

For sequencing trajectories in time instead, see CompositeTrajectory.

All of the underlying Trajectory objects must have the same %start_time() and %end_time().

When constructed with rowwise set to true, all of the underlying Trajectory objects must have the same number of %cols() and the value() matrix will be the vstack of the the trajectories in the order they were added.

When constructed with rowwise set to false, all of the underlying Trajectory objects must have the same number of %rows() and the value() matrix will be the hstack of the the trajectories in the order they were added.

__init__(self: pydrake.trajectories.StackedTrajectory_[AutoDiffXd], rowwise: bool = True) → None

Creates an empty trajectory.

Parameter rowwise:: governs the stacking order

Append(self: pydrake.trajectories.StackedTrajectory_[AutoDiffXd], arg0: pydrake.trajectories.Trajectory_[AutoDiffXd]) → None

Stacks another sub-Trajectory onto this. Refer to the class overview documentation for details.

Raises: RuntimeError if the matrix dimension is incompatible. –

Clone(self: pydrake.trajectories.StackedTrajectory_[AutoDiffXd]) → pydrake.trajectories.Trajectory_[AutoDiffXd]

class pydrake.trajectories.StackedTrajectory_[Expression]

Bases: pydrake.trajectories.Trajectory_[Expression]

A StackedTrajectory stacks the values from one or more underlying Trajectory objects into a single Trajectory, without changing the %start_time() or %end_time().

For sequencing trajectories in time instead, see CompositeTrajectory.

All of the underlying Trajectory objects must have the same %start_time() and %end_time().

When constructed with rowwise set to true, all of the underlying Trajectory objects must have the same number of %cols() and the value() matrix will be the vstack of the the trajectories in the order they were added.

When constructed with rowwise set to false, all of the underlying Trajectory objects must have the same number of %rows() and the value() matrix will be the hstack of the the trajectories in the order they were added.

__init__(self: pydrake.trajectories.StackedTrajectory_[Expression], rowwise: bool = True) → None

Creates an empty trajectory.

Parameter rowwise:: governs the stacking order

Append(self: pydrake.trajectories.StackedTrajectory_[Expression], arg0: pydrake.trajectories.Trajectory_[Expression]) → None

Stacks another sub-Trajectory onto this. Refer to the class overview documentation for details.

Raises: RuntimeError if the matrix dimension is incompatible. –

Clone(self: pydrake.trajectories.StackedTrajectory_[Expression]) → pydrake.trajectories.Trajectory_[Expression]

class pydrake.trajectories.Trajectory

A Trajectory represents a time-varying matrix, indexed by a single scalar time.

Note

This class is templated; see Trajectory_ for the list of instantiations.

__init__(self: pydrake.trajectories.Trajectory) → None

cols(self: pydrake.trajectories.Trajectory) → int

Returns: The number of columns in the matrix returned by value().

end_time(self: pydrake.trajectories.Trajectory) → float

EvalDerivative(self: pydrake.trajectories.Trajectory, t: float, derivative_order: int = 1) → numpy.ndarray[numpy.float64[m, n]]

Evaluates the derivative of this at the given time t. Returns the nth derivative, where n is the value of derivative_order.

Raises: RuntimeError if derivative_order is negative. –

has_derivative(self: pydrake.trajectories.Trajectory) → bool: Returns true iff the Trajectory provides and implementation for EvalDerivative() and MakeDerivative(). The derivative need not be continuous, but should return a result for all t for which value(t) returns a result.

MakeDerivative(self: pydrake.trajectories.Trajectory, derivative_order: int = 1) → pydrake.trajectories.Trajectory

Takes the derivative of this Trajectory.

Parameter derivative_order:: The number of times to take the derivative before returning.

Returns: The nth derivative of this object.
Raises: RuntimeError if derivative_order is negative. –

rows(self: pydrake.trajectories.Trajectory) → int

Returns: The number of rows in the matrix returned by value().

start_time(self: pydrake.trajectories.Trajectory) → float

value(self: pydrake.trajectories.Trajectory, t: float) → numpy.ndarray[numpy.float64[m, n]]

Evaluates the trajectory at the given time t.

Parameter t:: The time at which to evaluate the trajectory.

Returns: The matrix of evaluated values.

vector_values(self: pydrake.trajectories.Trajectory, t: list[float]) → numpy.ndarray[numpy.float64[m, n]]

If cols()==1, then evaluates the trajectory at each time t, and returns the results as a Matrix with the ith column corresponding to the ith time. Otherwise, if rows()==1, then evaluates the trajectory at each time t, and returns the results as a Matrix with the ith row corresponding to the ith time.

Raises: RuntimeError if both cols and rows are not equal to 1. –

template pydrake.trajectories.Trajectory_: Instantiations: Trajectory_[float], Trajectory_[AutoDiffXd], Trajectory_[Expression]

class pydrake.trajectories.Trajectory_[AutoDiffXd]

A Trajectory represents a time-varying matrix, indexed by a single scalar time.

__init__(self: pydrake.trajectories.Trajectory_[AutoDiffXd]) → None

cols(self: pydrake.trajectories.Trajectory_[AutoDiffXd]) → int

Returns: The number of columns in the matrix returned by value().

end_time(self: pydrake.trajectories.Trajectory_[AutoDiffXd]) → pydrake.autodiffutils.AutoDiffXd

EvalDerivative(self: pydrake.trajectories.Trajectory_[AutoDiffXd], t: pydrake.autodiffutils.AutoDiffXd, derivative_order: int = 1) → numpy.ndarray[object[m, n]]

Evaluates the derivative of this at the given time t. Returns the nth derivative, where n is the value of derivative_order.

Raises: RuntimeError if derivative_order is negative. –

has_derivative(self: pydrake.trajectories.Trajectory_[AutoDiffXd]) → bool: Returns true iff the Trajectory provides and implementation for EvalDerivative() and MakeDerivative(). The derivative need not be continuous, but should return a result for all t for which value(t) returns a result.

MakeDerivative(self: pydrake.trajectories.Trajectory_[AutoDiffXd], derivative_order: int = 1) → pydrake.trajectories.Trajectory_[AutoDiffXd]

Takes the derivative of this Trajectory.

Parameter derivative_order:: The number of times to take the derivative before returning.

Returns: The nth derivative of this object.
Raises: RuntimeError if derivative_order is negative. –

rows(self: pydrake.trajectories.Trajectory_[AutoDiffXd]) → int

Returns: The number of rows in the matrix returned by value().

start_time(self: pydrake.trajectories.Trajectory_[AutoDiffXd]) → pydrake.autodiffutils.AutoDiffXd

value(self: pydrake.trajectories.Trajectory_[AutoDiffXd], t: pydrake.autodiffutils.AutoDiffXd) → numpy.ndarray[object[m, n]]

Evaluates the trajectory at the given time t.

Parameter t:: The time at which to evaluate the trajectory.

Returns: The matrix of evaluated values.

vector_values(self: pydrake.trajectories.Trajectory_[AutoDiffXd], t: list[pydrake.autodiffutils.AutoDiffXd]) → numpy.ndarray[object[m, n]]

If cols()==1, then evaluates the trajectory at each time t, and returns the results as a Matrix with the ith column corresponding to the ith time. Otherwise, if rows()==1, then evaluates the trajectory at each time t, and returns the results as a Matrix with the ith row corresponding to the ith time.

Raises: RuntimeError if both cols and rows are not equal to 1. –

class pydrake.trajectories.Trajectory_[Expression]

A Trajectory represents a time-varying matrix, indexed by a single scalar time.

__init__(self: pydrake.trajectories.Trajectory_[Expression]) → None

cols(self: pydrake.trajectories.Trajectory_[Expression]) → int

Returns: The number of columns in the matrix returned by value().

end_time(self: pydrake.trajectories.Trajectory_[Expression]) → pydrake.symbolic.Expression

EvalDerivative(self: pydrake.trajectories.Trajectory_[Expression], t: pydrake.symbolic.Expression, derivative_order: int = 1) → numpy.ndarray[object[m, n]]

Evaluates the derivative of this at the given time t. Returns the nth derivative, where n is the value of derivative_order.

Raises: RuntimeError if derivative_order is negative. –

has_derivative(self: pydrake.trajectories.Trajectory_[Expression]) → bool: Returns true iff the Trajectory provides and implementation for EvalDerivative() and MakeDerivative(). The derivative need not be continuous, but should return a result for all t for which value(t) returns a result.

MakeDerivative(self: pydrake.trajectories.Trajectory_[Expression], derivative_order: int = 1) → pydrake.trajectories.Trajectory_[Expression]

Takes the derivative of this Trajectory.

Parameter derivative_order:: The number of times to take the derivative before returning.

Returns: The nth derivative of this object.
Raises: RuntimeError if derivative_order is negative. –

rows(self: pydrake.trajectories.Trajectory_[Expression]) → int

Returns: The number of rows in the matrix returned by value().

start_time(self: pydrake.trajectories.Trajectory_[Expression]) → pydrake.symbolic.Expression

value(self: pydrake.trajectories.Trajectory_[Expression], t: pydrake.symbolic.Expression) → numpy.ndarray[object[m, n]]

Evaluates the trajectory at the given time t.

Parameter t:: The time at which to evaluate the trajectory.

Returns: The matrix of evaluated values.

vector_values(self: pydrake.trajectories.Trajectory_[Expression], t: list[pydrake.symbolic.Expression]) → numpy.ndarray[object[m, n]]

If cols()==1, then evaluates the trajectory at each time t, and returns the results as a Matrix with the ith column corresponding to the ith time. Otherwise, if rows()==1, then evaluates the trajectory at each time t, and returns the results as a Matrix with the ith row corresponding to the ith time.

Raises: RuntimeError if both cols and rows are not equal to 1. –