Drake

## Detailed Description

Implementations of controllers that operate as Systems in a block diagram.

Algorithms that synthesize controllers are located in Feedback Control Design.

## Classes

class  InverseDynamicsController< T >
A state feedback controller that uses a PidController to generate desired accelerations, which are then converted into torques using InverseDynamics. More...

class  LinearModelPredictiveController< T >
Implements a basic Model Predictive Controller that linearizes the system about an equilibrium condition and regulates to the same point by solving an optimal control problem over a finite time horizon. More...

class  PidControlledSystem< T >
A system that encapsulates a PidController and a controlled System (a.k.a the "plant"). More...

class  PidController< T >
Implements the PID controller. More...

## Functions

std::pair< std::unique_ptr< BarycentricMeshSystem< double > >, Eigen::RowVectorXd > FittedValueIteration (Simulator< double > *simulator, const std::function< double(const Context< double > &context)> &cost_function, const math::BarycentricMesh< double >::MeshGrid &state_grid, const math::BarycentricMesh< double >::MeshGrid &input_grid, double timestep, const DynamicProgrammingOptions &options=DynamicProgrammingOptions())
Implements Fitted Value Iteration on a (triangulated) Barycentric Mesh, which designs a state-feedback policy to minimize the infinite-horizon cost ∑ γⁿ g(x[n],u[n]), where γ is the discount factor in options. More...

Eigen::VectorXd LinearProgrammingApproximateDynamicProgramming (Simulator< double > *simulator, const std::function< double(const Context< double > &context)> &cost_function, const std::function< symbolic::Expression(const Eigen::Ref< const Eigen::VectorXd > &state, const VectorX< symbolic::Variable > &parameters)> &linearly_parameterized_cost_to_go_function, int num_parameters, const Eigen::Ref< const Eigen::MatrixXd > &state_samples, const Eigen::Ref< const Eigen::MatrixXd > &input_samples, double timestep, const DynamicProgrammingOptions &options=DynamicProgrammingOptions())
Implements the Linear Programming approach to approximate dynamic programming. More...

FiniteHorizonLinearQuadraticRegulatorResult FiniteHorizonLinearQuadraticRegulator (const System< double > &system, const Context< double > &context, double t0, double tf, const Eigen::Ref< const Eigen::MatrixXd > &Q, const Eigen::Ref< const Eigen::MatrixXd > &R, const FiniteHorizonLinearQuadraticRegulatorOptions &options=FiniteHorizonLinearQuadraticRegulatorOptions())
Solves the differential Riccati equation to compute the optimal controller and optimal cost-to-go for the finite-horizon linear quadratic regulator: More...

std::unique_ptr< System< double > > MakeFiniteHorizonLinearQuadraticRegulator (const System< double > &system, const Context< double > &context, double t0, double tf, const Eigen::Ref< const Eigen::MatrixXd > &Q, const Eigen::Ref< const Eigen::MatrixXd > &R, const FiniteHorizonLinearQuadraticRegulatorOptions &options=FiniteHorizonLinearQuadraticRegulatorOptions())
Variant of FiniteHorizonLinearQuadraticRegulator that returns a System implementing the regulator (controller) as a System, with a single "plant_state" input for the estimated plant state, and a single "control" output for the regulator control output. More...

LinearQuadraticRegulatorResult LinearQuadraticRegulator (const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::MatrixXd > &B, const Eigen::Ref< const Eigen::MatrixXd > &Q, const Eigen::Ref< const Eigen::MatrixXd > &R, const Eigen::Ref< const Eigen::MatrixXd > &N=Eigen::Matrix< double, 0, 0 >::Zero())
Computes the optimal feedback controller, u=-Kx, and the optimal cost-to-go J = x'Sx for the problem: More...

LinearQuadraticRegulatorResult DiscreteTimeLinearQuadraticRegulator (const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::MatrixXd > &B, const Eigen::Ref< const Eigen::MatrixXd > &Q, const Eigen::Ref< const Eigen::MatrixXd > &R)
Computes the optimal feedback controller, u=-Kx, and the optimal cost-to-go J = x'Sx for the problem: More...

std::unique_ptr< LinearSystem< double > > LinearQuadraticRegulator (const LinearSystem< double > &system, const Eigen::Ref< const Eigen::MatrixXd > &Q, const Eigen::Ref< const Eigen::MatrixXd > &R, const Eigen::Ref< const Eigen::MatrixXd > &N=Eigen::Matrix< double, 0, 0 >::Zero())
Creates a system that implements the optimal time-invariant linear quadratic regulator (LQR). More...

std::unique_ptr< AffineSystem< double > > LinearQuadraticRegulator (const System< double > &system, const Context< double > &context, const Eigen::Ref< const Eigen::MatrixXd > &Q, const Eigen::Ref< const Eigen::MatrixXd > &R, const Eigen::Ref< const Eigen::MatrixXd > &N=Eigen::Matrix< double, 0, 0 >::Zero(), int input_port_index=0)
Linearizes the System around the specified Context, computes the optimal time-invariant linear quadratic regulator (LQR), and returns a System which implements that regulator in the original System's coordinates. More...

Eigen::MatrixXd ControllabilityMatrix (const LinearSystem< double > &sys)
Returns the controllability matrix: R = [B, AB, ..., A^{n-1}B]. More...

bool IsControllable (const LinearSystem< double > &sys, std::optional< double > threshold=std::nullopt)
Returns true iff the controllability matrix is full row rank. More...

## ◆ ControllabilityMatrix()

 Eigen::MatrixXd drake::systems::ControllabilityMatrix ( const LinearSystem< double > & sys )

Returns the controllability matrix: R = [B, AB, ..., A^{n-1}B].

 LinearQuadraticRegulatorResult drake::systems::controllers::DiscreteTimeLinearQuadraticRegulator ( const Eigen::Ref< const Eigen::MatrixXd > & A, const Eigen::Ref< const Eigen::MatrixXd > & B, const Eigen::Ref< const Eigen::MatrixXd > & Q, const Eigen::Ref< const Eigen::MatrixXd > & R )

Computes the optimal feedback controller, u=-Kx, and the optimal cost-to-go J = x'Sx for the problem:

$x[n+1] = Ax[n] + Bu[n]$

$\min_u \sum_0^\infty x'Qx + u'Ru$

Parameters
 A The state-space dynamics matrix of size num_states x num_states. B The state-space input matrix of size num_states x num_inputs. Q A symmetric positive semi-definite cost matrix of size num_states x num_states. R A symmetric positive definite cost matrix of size num_inputs x num_inputs.
Returns
A structure that contains the optimal feedback gain K and the quadratic cost term S. The optimal feedback control is u = -Kx;
Exceptions
 std::runtime_error if R is not positive definite.

 FiniteHorizonLinearQuadraticRegulatorResult drake::systems::controllers::FiniteHorizonLinearQuadraticRegulator ( const System< double > & system, const Context< double > & context, double t0, double tf, const Eigen::Ref< const Eigen::MatrixXd > & Q, const Eigen::Ref< const Eigen::MatrixXd > & R, const FiniteHorizonLinearQuadraticRegulatorOptions & options = FiniteHorizonLinearQuadraticRegulatorOptions() )

Solves the differential Riccati equation to compute the optimal controller and optimal cost-to-go for the finite-horizon linear quadratic regulator:

$\min_u (x(t_f)-x_d(t_f))'Q_f(x(t_f)-x_d(t_f)) + \int_{t_0}^{t_f} (x(t)-x_d(t))'Q(x(t)-x_d(t)) dt + \int_{t_0}^{t_f} (u(t)-u_d(t))'R(u(t)-u_d(t)) dt + \int_{t_0}^{t_f} 2(x(t)-x_d(t))'N(u(t)-u_d(t)) dt \\ \text{s.t. } \dot{x} - \dot{x}_0(t) = A(t)(x(t) - x_0(t)) + B(t)(u(t) - u_0(t)) + c(t)$

where A(t), B(t), and c(t) are taken from the gradients of the continuous-time dynamics ẋ = f(t,x,u), as A(t) = dfdx(t, x0(t), u0(t)), B(t) = dfdu(t, x0(t), u0(t)), and c(t) = f(t, x0(t), u0(t)) - ẋ0(t). x0(t) and u0(t) can be specified in options, otherwise are taken to be constant trajectories with values given by context.

Parameters
 system a System representing the plant. context a Context used to pass the default input, state, and parameters. Note: Use options to specify time-varying nominal state and/or input trajectories. t0 is the initial time. tf is the final time (with tf > t0). Q is nxn positive semi-definite. R is mxm positive definite. options is the optional FiniteHorizonLinearQuadraticRegulatorOptions.
Precondition
system must be a System<double> with (only) n continuous state variables and m inputs. It must be convertable to System<AutoDiffXd>.
Note
Support for difference-equation systems (
System<T>::IsDifferenceEquationSystem()) by solving the differential Riccati equation and richer specification of the objective are anticipated (they are listed in the code as TODOs).

## ◆ FittedValueIteration()

 std::pair >, Eigen::RowVectorXd> drake::systems::controllers::FittedValueIteration ( Simulator< double > * simulator, const std::function< double(const Context< double > &context)> & cost_function, const math::BarycentricMesh< double >::MeshGrid & state_grid, const math::BarycentricMesh< double >::MeshGrid & input_grid, double timestep, const DynamicProgrammingOptions & options = DynamicProgrammingOptions() )

Implements Fitted Value Iteration on a (triangulated) Barycentric Mesh, which designs a state-feedback policy to minimize the infinite-horizon cost ∑ γⁿ g(x[n],u[n]), where γ is the discount factor in options.

For background, and a description of this algorithm, see http://underactuated.csail.mit.edu/underactuated.html?chapter=dp . It currently requires that the system to be optimized has only continuous state and it is assumed to be time invariant. This code makes a discrete-time approximation (using timestep) for the value iteration update.

Parameters
 simulator contains the reference to the System being optimized and to a Context for that system, which may contain non-default Parameters, etc. The simulator is run for timestep seconds from every point on the mesh in order to approximate the dynamics; all of the simulation parameters (integrator, etc) are relevant during that evaluation. cost_function is the continuous-time instantaneous cost. This implementation of the discrete-time formulation above uses the approximation g(x,u) = timestep*cost_function(x,u). state_grid defines the mesh on the state space used to represent the cost-to-go function and the resulting policy. input_grid defines the discrete action space used in the value iteration update. timestep a time in seconds used for the discrete-time approximation. options optional DynamicProgrammingOptions structure.
Returns
a std::pair containing the resulting policy, implemented as a BarycentricMeshSystem, and the RowVectorXd J that defines the expected cost-to-go on a BarycentricMesh using state_grid. The policy has a single vector input (which is the continuous state of the system passed in through simulator) and a single vector output (which is the input of the system passed in through simulator).

## ◆ IsControllable()

 bool drake::systems::IsControllable ( const LinearSystem< double > & sys, std::optional< double > threshold = std::nullopt )

Returns true iff the controllability matrix is full row rank.

## ◆ LinearProgrammingApproximateDynamicProgramming()

 Eigen::VectorXd drake::systems::controllers::LinearProgrammingApproximateDynamicProgramming ( Simulator< double > * simulator, const std::function< double(const Context< double > &context)> & cost_function, const std::function< symbolic::Expression(const Eigen::Ref< const Eigen::VectorXd > &state, const VectorX< symbolic::Variable > ¶meters)> & linearly_parameterized_cost_to_go_function, int num_parameters, const Eigen::Ref< const Eigen::MatrixXd > & state_samples, const Eigen::Ref< const Eigen::MatrixXd > & input_samples, double timestep, const DynamicProgrammingOptions & options = DynamicProgrammingOptions() )

Implements the Linear Programming approach to approximate dynamic programming.

It optimizes the linear program

maximize ∑ Jₚ(x). subject to ∀x, ∀u, Jₚ(x) ≤ g(x,u) + γJₚ(f(x,u)),

where g(x,u) is the one-step cost, Jₚ(x) is a (linearly) parameterized cost-to-go function with parameter vector p, and γ is the discount factor in options.

For background, and a description of this algorithm, see http://underactuated.csail.mit.edu/underactuated.html?chapter=dp . It currently requires that the system to be optimized has only continuous state and it is assumed to be time invariant. This code makes a discrete-time approximation (using timestep) for the value iteration update.

Parameters
 simulator contains the reference to the System being optimized and to a Context for that system, which may contain non-default Parameters, etc. The simulator is run for timestep seconds from every pair of input/state sample points in order to approximate the dynamics; all of the simulation parameters (integrator, etc) are relevant during that evaluation. cost_function is the continuous-time instantaneous cost. This implementation of the discrete-time formulation above uses the approximation g(x,u) = timestep*cost_function(x,u). linearly_parameterized_cost_to_go_function must define a function to approximate the cost-to-go, which takes the state vector as the first input and the parameter vector as the second input. This can be any function of the form Jₚ(x) = ∑ pᵢ φᵢ(x). This algorithm will pass in a VectorX of symbolic::Variable in order to set up the linear program. state_samples is a list of sample states (one per column) at which to apply the optimization constraints and the objective. input_samples is a list of inputs (one per column) which are evaluated at every sample point. timestep a time in seconds used for the discrete-time approximation. options optional DynamicProgrammingOptions structure.
Returns
params the VectorXd of parameters that optimizes the supplied cost-to-go function.

 LinearQuadraticRegulatorResult drake::systems::controllers::LinearQuadraticRegulator ( const Eigen::Ref< const Eigen::MatrixXd > & A, const Eigen::Ref< const Eigen::MatrixXd > & B, const Eigen::Ref< const Eigen::MatrixXd > & Q, const Eigen::Ref< const Eigen::MatrixXd > & R, const Eigen::Ref< const Eigen::MatrixXd > & N = Eigen::Matrix< double, 0, 0 >::Zero() )

Computes the optimal feedback controller, u=-Kx, and the optimal cost-to-go J = x'Sx for the problem:

$\dot{x} = Ax + Bu$

$\min_u \int_0^\infty x'Qx + u'Ru + 2x'Nu dt$

Parameters
 A The state-space dynamics matrix of size num_states x num_states. B The state-space input matrix of size num_states x num_inputs. Q A symmetric positive semi-definite cost matrix of size num_states x num_states. R A symmetric positive definite cost matrix of size num_inputs x num_inputs. N A cost matrix of size num_states x num_inputs. If the matrix is zero-sized, N will be treated as a num_states x num_inputs zero matrix.
Returns
A structure that contains the optimal feedback gain K and the quadratic cost term S. The optimal feedback control is u = -Kx;
Exceptions
 std::runtime_error if R is not positive definite.

 std::unique_ptr > drake::systems::controllers::LinearQuadraticRegulator ( const LinearSystem< double > & system, const Eigen::Ref< const Eigen::MatrixXd > & Q, const Eigen::Ref< const Eigen::MatrixXd > & R, const Eigen::Ref< const Eigen::MatrixXd > & N = Eigen::Matrix< double, 0, 0 >::Zero() )

Creates a system that implements the optimal time-invariant linear quadratic regulator (LQR).

If system is a continuous-time system, then solves the continuous-time LQR problem:

$\min_u \int_0^\infty x^T(t)Qx(t) + u^T(t)Ru(t) dt.$

If system is a discrete-time system, then solves the discrete-time LQR problem:

$\min_u \sum_0^\infty x^T[n]Qx[n] + u^T[n]Ru[n].$

Parameters
 system The System to be controlled. Q A symmetric positive semi-definite cost matrix of size num_states x num_states. R A symmetric positive definite cost matrix of size num_inputs x num_inputs. N A cost matrix of size num_states x num_inputs.
Returns
A system implementing the optimal controller in the original system coordinates.
Exceptions
 std::runtime_error if R is not positive definite.

 std::unique_ptr > drake::systems::controllers::LinearQuadraticRegulator ( const System< double > & system, const Context< double > & context, const Eigen::Ref< const Eigen::MatrixXd > & Q, const Eigen::Ref< const Eigen::MatrixXd > & R, const Eigen::Ref< const Eigen::MatrixXd > & N = Eigen::Matrix< double, 0, 0 >::Zero(), int input_port_index = 0 )

Linearizes the System around the specified Context, computes the optimal time-invariant linear quadratic regulator (LQR), and returns a System which implements that regulator in the original System's coordinates.

If system is a continuous-time system, then solves the continuous-time LQR problem:

$\min_u \int_0^\infty (x-x_0)^TQ(x-x_0) + (u-u_0)^TR(u-u_0) dt.$

If system is a discrete-time system, then solves the discrete-time LQR problem:

$\min_u \sum_0^\infty (x-x_0)^TQ(x-x_0) + (u-u_0)^TR(u-u_0),$

where $$x_0$$ is the nominal state and $$u_0$$ is the nominal input. The system is considered discrete if it has a single discrete state vector and a single unique periodic update event declared.

Parameters
 system The System to be controlled. context Defines the desired state and control input to regulate the system to. Note that this state/input must be an equilibrium point of the system. See drake::systems::Linearize for more details. Q A symmetric positive semi-definite cost matrix of size num_states x num_states. R A symmetric positive definite cost matrix of size num_inputs x num_inputs. N A cost matrix of size num_states x num_inputs. If the matrix is zero-sized, N will be treated as a num_states x num_inputs zero matrix. int_port_index The index of the input port to linearize around.
Returns
A system implementing the optimal controller in the original system coordinates.
Exceptions
 std::runtime_error if R is not positive definite.
 std::unique_ptr > drake::systems::controllers::MakeFiniteHorizonLinearQuadraticRegulator ( const System< double > & system, const Context< double > & context, double t0, double tf, const Eigen::Ref< const Eigen::MatrixXd > & Q, const Eigen::Ref< const Eigen::MatrixXd > & R, const FiniteHorizonLinearQuadraticRegulatorOptions & options = FiniteHorizonLinearQuadraticRegulatorOptions() )