Trajectory Tracking

Definition

Trajectory tracking is the control problem of driving a system’s state to follow a time-varying reference ${\mathbf{q}}_d(t)$ (as opposed to a fixed set-point), so that the tracking error $\tilde{\mathbf{q}}=\mathbf{q}-{\mathbf{q}}_d\to 0$ (or to a small ultimate-bound neighborhood). For a space manipulator the generalized coordinate $\mathbf{q}$ stacks the actuated base pose and the arm joint angles, and the closed loop must respect the strong base–arm dynamic coupling (dynamic_coupling): any arm motion disturbs the base and vice-versa (yao2021adaptive). The regime matters. Our system is free-flying (fully-actuated base), so every generalized DOF has an actuator and the problem reduces to tracking control of a fully-actuated Euler–Lagrange system; the more common free-floating regime (ye2019research) lacks base actuation, so reference motions must additionally honor momentum conservation. Because the tracking-error dynamics of such mechanical systems decompose into a cascade of a stable nominal loop perturbed by a vanishing signal, asymptotic tracking is certified by cascaded-systems Lyapunov theory (panteley1998global, panteley2001growth).

Key Equations

Symbols per notation.md.

Free-flying manipulator plant (fully-actuated Euler–Lagrange form; $\mathbf{M}$ SPD, $\dot{\mathbf{M}}-2\mathbf{C}$ skew-symmetric — yao2021adaptive Eq. 1 with Properties 1–2; the form below drops the saturation, i.e. the source’s unsaturated Eq. 4 $\mathbf{\tau }+\Delta \mathbf{\tau }+\mathbf{d}$ with $\Delta \mathbf{\tau }=0$ — the source’s Eq. 1 carries $\mathrm{sat}(\mathbf{\tau })$ in place of $\mathbf{\tau }$):
$\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}=\mathbf{\tau }+\mathbf{d}$

Tracking objective — generalized position / velocity errors (yao2021adaptive Eqs. 11–12; here $\tilde{\mathbf{q}}\equiv {\mathbf{e}}_1$ instantiates the canonical $\widetilde{(\cdot )}$ error convention, and $\mathbf{\mu }$ is the source’s auxiliary (stabilizing-reference) velocity — so ${\mathbf{e}}_2\equiv \dot{\mathbf{q}}-\mathbf{\mu }$ is the velocity-error variable, which coincides with $\dot{\tilde{\mathbf{q}}}$ only when $\mathbf{\mu }={\dot{\mathbf{q}}}_d$):
$\tilde{\mathbf{q}}=\mathbf{q}-{\mathbf{q}}_d,{\mathbf{e}}_2=\dot{\mathbf{q}}-\mathbf{\mu },\tilde{\mathbf{q}}(t)\to 0(\text{or a small ultimate bound}).$

Stability mechanism — the error system is read as a cascade ${\dot{x}}_1=f_1(t,x_1,x_2),{\dot{x}}_2=f_2(t,x_2)$; if the unperturbed loop is UGAS and the perturbing channel is integrable (or satisfies the growth-rate condition), the interconnection is UGAS (panteley1998global Thm. 2; panteley2001growth):
${\dot{x}}_1=f_1(t,x_1)+g(t,x_1,x_2)x_2,{\dot{x}}_2=f_2(t,x_2)\text{UGAS}\Rightarrow \text{cascade UGAS}.$

Source Support

• yao2021adaptive — primary: adaptive trajectory-tracking control of an attitude-controlled free-flying space manipulator; gives the plant model, the position/velocity error definitions, and a barrier-Lyapunov + RBFNN + high-gain-observer design proving semi-global uniform ultimate boundedness under uncertainty, disturbance, and actuator saturation.
• ye2019research — support / regime contrast: joint-space trajectory tracking of a free-floating redundant space robot via adaptive robust control coupled with reaction-null-space planning; supplies the same error/sliding-variable structure ($\mathbf{r}=\dot{\tilde{\mathbf{q}}}+\mathbf{\Lambda }\tilde{\mathbf{q}}$) but in the unactuated-base regime.
• panteley1998global — theory: sufficient Lyapunov conditions for (global) uniform asymptotic stability of time-varying cascaded systems — the certificate that a tracking error decomposed into a cascade converges.
• panteley2001growth — theory: relaxes the integrability requirement to growth-rate conditions on the Lyapunov function; explicitly motivated by trajectory-tracking control of non-autonomous (time-varying-reference) systems.

Related Topics

• task_space_error_dynamics — the end-effector-level error model whose convergence is tracking when the reference is specified in the task (Cartesian) space rather than joint space.
• cascaded_systems — the structural decomposition (Panteley–Loría) used to certify tracking-error convergence; the bridge from these sources to a stability proof.
• lyapunov_stability — the direct method (incl. barrier-Lyapunov functions) supplying the convergence/ultimate-boundedness certificate for the tracking loop.
• trajectory_optimization — generates the reference ${\mathbf{q}}_d(t)$ that this controller is tasked to follow (planning vs. tracking split).
• closed_loop_inverse_kinematics — the velocity/kinematic-level realization of tracking, mapping a desired end-effector path to joint rates with error feedback.
• Also relevant: prescribed_performance_control and barrier_lyapunov_function (transient/steady-state bounds on the tracking error in yao2021adaptive), reaction_null_space (the free-floating coupling constraint in ye2019research), and free_flying_vs_free_floating (the regime distinction that changes the actuation and the constraints).

Open Questions

• yao2021adaptive models the attitude-controlled free-flying mode: its generalized coordinate $\mathbf{q}$ contains the base attitude and joint angles but not base translation. Does the same error model and BLF/RBFNN proof carry over unchanged to our fully 6-DOF actuated base (translation actuated too), or does the added translational loop change the coupling structure and the bounds?
• The cascade certificates (panteley1998global, panteley2001growth) give global/uniform asymptotic stability for the ideal error system, whereas yao2021adaptive attains only semi-global uniform ultimate boundedness under saturation and NN-approximation error. What reference aggressiveness / disturbance bound keeps the ultimate-bound neighborhood inside the prescribed-performance funnel for our system?
• ye2019research achieves tracking while enforcing reaction-null-space (base-disturbance-minimizing) constraints — a free-floating necessity. With a fully-actuated base we can instead reject base disturbance directly: is reactionless tracking still worthwhile (e.g. to spare base actuator effort / momentum) for a free-flying inspector?

Implementation (sims wiki)

External — into the code wiki via the sims_wiki/ symlink (resolves in Obsidian, not GitHub).

• ee_guidance — closed-loop end-effector tracking error dynamics + feedforward.