Control & Motion Planning

Kinematics and dynamics tell us what a robot can do; control tells us how to make it do that. A controller takes a desired trajectory and a model of the system, and produces the inputs (joint torques, motor voltages) that drive the real system toward the desired behavior.

The classical tool for this is state-space control: represent the system as a set of first-order ODEs in a state vector $x$, linearize about an operating point to get an LTI model $\dot{x}=Ax+Bu$, and design feedback laws using eigenvalue and rank conditions on $A$ and $B$.

Notes

• 7.1. State-Space Representation — States, inputs, outputs; $\dot{x}=f(x,u,t)$; LTI form $\dot{x}=Ax+Bu$; continuous vs discrete time
• 7.2. Linearization — Taylor expansion of $f$ and $h$ about an equilibrium to produce $(A,B,C,D)$ from the Jacobians of the dynamics
• 7.3. Solving LTI Systems — The state transition matrix $e^{At}$, homogeneous and particular solutions, eigendecomposition
• 7.4. Phase Portraits — Vector fields of $\dot{x}=Ax$; classification of 2D systems (node, saddle, focus, center)
• 7.5. Stability — Internal (eigenvalue) stability, BIBO stability, location of eigenvalues in the complex plane
• 7.6. State Feedback — Closed-loop dynamics $A_c=A-BK$; pole placement via $u=-Kx+Fv$
• 7.7. Controllability — Controllability matrix $\mathcal{C}=[BAB{A}^{2}B\cdots A^{n-1}B]$, rank condition, reachability