osqp-eigen Documentation

Since osqp-eigen only handles standard Quadratic Programming (QP) problems, users implementing Model Predictive Control (MPC) must manually cast the MPC formulation into a QP form.

An MPC problem involving a linear system tracking a reference state $x_r$ with state/input constraints can be mapped to the standard QP form:

Minimize: $\frac{1}{2} x^T P x + q^T x$ Subject to: $l \leq A_c x \leq u$

Where:

• Hessian ($P$): A diagonal matrix of the cost matrices (e.g., $Q$ and $R$).
• Gradient ($q$): Derived from the reference state $x_r$ and cost matrices.
• Constraint Matrix ($A_c$): Encodes the system dynamics ($x_{k+1} = Ax_k + Bu_k$) and initial state.
• Bounds ($l, u$): Encodes state/input limits and the initial state constraint.

Reference implementations for these conversion functions are available in the repository's example code.

// Reference implementation functions for casting MPC to QP
castMPCToQPHessian(Q, R, mpcWindow, hessian);
castMPCToQPGradient(Q, xRef, mpcWindow, gradient);
castMPCToQPConstraintMatrix(a, b, mpcWindow, linearMatrix);
castMPCToQPConstraintVectors(xMax, xMin, uMax, uMin, x0, mpcWindow, lowerBound, upperBound);

To solve a QP problem using osqp-eigen, follow these steps:

1. Instantiate the solver: Create an OsqpEigen::Solver instance.
2. Configure settings: Use solver.settings()->set<Setting>() to modify default behaviors (e.g., enabling warm start).
3. Set problem data: Define the number of variables/constraints and provide the Hessian, gradient, constraint matrix, and bounds using the solver.data()->set... methods. These methods return true on success.
4. Initialize: Call solver.initSolver() to prepare the underlying OSQP structure.
5. Solve: Call solver.solveProblem() and check the return flag against OsqpEigen::ErrorExitFlag::NoError.
6. Retrieve solution: Use solver.getSolution() to get the resulting Eigen::VectorXd.

// 1. Instantiate
OsqpEigen::Solver solver;

// 2. Configure settings
solver.settings()->setWarmStart(true);

// 3. Set data
solver.data()->setNumberOfVariables(numberOfVariable);
solver.data()->setNumberOfConstraints(numberOfConstraints);
if(!solver.data()->setHessianMatrix(hessian)) return 1;
if(!solver.data()->setGradient(gradient)) return 1;
if(!solver.data()->setLinearConstraintMatrix(linearMatrix)) return 1;
if(!solver.data()->setLowerBound(lowerBound)) return 1;
if(!solver.data()->setUpperBound(upperBound)) return 1;

// 4. Initialize
if(!solver.initSolver()) return 1;

// 5. Solve
if(solver.solveProblem() != OsqpEigen::ErrorExitFlag::NoError) return 1;

// 6. Get solution
Eigen::VectorXd QPSolution = solver.getSolution();