Summary.
The chapter studies fixed-base robots whose dynamics exhibit coupled nonlinearity and quadratic velocity terms that can cause finite-time divergence; the antidote is the passivity of the natural dynamics, expressed through the skew-symmetry identity \dot{M} =C+C^T. After surveying control objectives on a 1-DOF double integrator, four representative controllers are analyzed via Lyapunov stability. Joint Compliance Control (JCC) imposes virtual stiffness and damping; with exact gravity compensation a positive-definite energy-like function yields \dot{V} = - \dot{\theta}^T D \dot{\theta} \le 0, upgraded to asymptotic stability by a LaSalle argument. Augmented PD (APD) extends JCC to tracking but yields only a negative-semidefinite \dot{V}. Inverse Dynamics Control (IDC) achieves exponential error dynamics with a perfect model but sacrifices passivity. Joint Impedance Control (JIC) reshapes apparent inertia using measured external torque but degrades under modeling error and friction.
The unresolved gaps — APD’s incomplete stability and IDC’s lost passivity — motivate the inverse Lyapunov method: instead of guessing a Lyapunov function, one posits a quadratic form and solves the resulting matrix differential Lyapunov equation (mDLE), converting an intractable partial differential inequality into a matrix ODE. Applying this, Passivity-based APD — which re-inject passivity via a reference velocity \dot{\theta}^{\text{ref}} = \dot{\theta}^{\text{des}} + \Phi e — and Passivity-based IDC are proved globally exponentially stable for ideal case, and their robustness against parametric uncertainty (recast as an extended disturbance free of acceleration and quadratic-velocity terms) is established.
The chapter then asks which stabilizing controller is best, leading to nonlinear optimal control: Pontryagin’s minimum principle and dynamic programming are presented, the latter yielding the HJB equation whose value function serves as a Lyapunov function. Using an inverse approach on the control-affine robot dynamics, the analytic solution to the inverse quadratic optimal control problem is obtained as a controller of pure PID form, independent of dynamic parameters. The same inverse strategy is carried to the disturbance-and-control-affine setting, deriving the HJI equation for \mathcal{L}_2-gain (\mathcal{H}_{\infty}) disturbance attenuation; because HJI shares the HJB structure, the very same reference-error PID feedback becomes the inverse \mathcal{H}_{\infty} sub-optimal controller. Robust stability is proved in the input-to-state stability (ISS) sense, a performance-limitation bound relates achievable tracking error to the \mathcal{L}_2-gain \gamma, and a practical square-linear tuning method and a decentralized \mathcal{H}_{\infty} controller follow. Finally, these results are generalized into the NRIC, a two-degree-of-freedom structure in which an inner-loop auxiliary input attenuates the deviation between real and nominal robots, so that any nominally stable outer-loop controller inherits \mathcal{H}_{\infty} robustness and ISS without sacrificing model nonlinearity.
Connection to subsequent chapters.
Chapter 9 relies directly on Chapter 7’s computational dynamics — RNE for inverse-dynamics torque computation and pRNE for the Christoffel-consistent C(\theta,\dot{\theta}) factorization that makes passivity injection and the extended-disturbance formulation rigorous — and on Chapter 8 for the desired joint trajectories. Its results are the template for Chapter 10, which lifts every controller to task space and to SO(3)/SE(3) tracking and impedance control. The NRIC framework in particular is reused in Chapter 10 as the unifying robustness layer for task-space motion and force control, making this chapter the analytical backbone of the book’s control theory.