Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

Robot Control: PID, Impedance, and Whole-Body Control

4 minute read

Published:

TL;DR: Robot control spans a hierarchy from low-level PID joint position loops (kHz) to high-level task-space controllers (100 Hz) to planning (1–10 Hz). Impedance control models the robot as a virtual spring-damper for compliant contact. Whole-body control uses quadratic programming to satisfy hierarchical task priorities. MPC optimises over a receding horizon for anticipatory behaviour.
Robot control for manipulation
Learned robot control for dexterous manipulation (Andrychowicz et al., 2019)

PID Control: The Universal Baseline

The PID (Proportional-Integral-Derivative) controller is the workhorse of industrial robot control. Given a position error \(e(t) = q_{\text{des}}(t) - q(t)\) between desired and actual joint angle, the control torque is:

$$u(t) = K_p\, e(t) + K_i \int_0^t e(\tau)\,d\tau + K_d\, \dot{e}(t)$$
  • Proportional term: restoring force proportional to error. High \(K_p\) gives fast response but can cause overshoot and oscillation.
  • Integral term: eliminates steady-state error by accumulating it over time. Can cause integral windup when the actuator saturates.
  • Derivative term: damping based on the rate of change of error. Reduces oscillation but amplifies sensor noise.

PID operates in joint space — one controller per joint. Cross-coupling between joints (due to inertia and Coriolis forces) is handled as a disturbance, which works for slow, light robots but breaks down for fast or heavy manipulation.

Computed torque control (a model-based approach) adds the inverse dynamics \(M(\mathbf{q})\ddot{\mathbf{q}} + C(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{g}(\mathbf{q})\) as a feedforward term, linearising and decoupling the system so that simple PID suffices in the error space.

Impedance Control: Compliant Interaction

Classic position control is unsuitable for contact-rich tasks: if the robot pushes against a rigid wall, position error accumulates and torques spike. Impedance control (Hogan 1985) models the robot end-effector as a virtual spring-damper system:

$$F = K(x_d - x) + D(\dot{x}_d - \dot{x}) + M_d(\ddot{x}_d - \ddot{x})$$

where \(K\) is the stiffness matrix, \(D\) is the damping matrix, and \(M_d\) is the desired inertia. In operational space, the controller applies forces \(F\) to drive the end-effector toward \(x_d\) while yielding compliantly to external forces.

Admittance control is the dual formulation: given measured forces, compute position corrections. This is preferred when the robot is stiff (high gear-ratio motors) and force control is not directly possible via joint torques.

Impedance control is essential for assembly tasks (peg-in-hole, screwing), human-robot interaction, and legged locomotion (contact with uncertain terrain).

Whole-Body Control

Humanoids and legged robots have many degrees of freedom (28–50 DOF) and must simultaneously satisfy multiple objectives: balance, end-effector task, joint limit avoidance, and contact force constraints. Whole-body control (WBC) formulates these as a hierarchical quadratic program (QP):

$$\min_{\ddot{\mathbf{q}}, \mathbf{f}} \;\|\text{Task}_1\|^2 + \epsilon\|\text{Task}_2\|^2 \quad \text{s.t.} \quad A\mathbf{x} \leq \mathbf{b}$$

Higher-priority tasks (balance) are solved exactly; lower-priority tasks (reaching) are solved in the null space of higher-priority constraints. The QP is solved in real time (1 kHz) using efficient active-set solvers. WBC was instrumental in enabling Atlas (Boston Dynamics) to perform dynamic manipulation and parkour.

Learning Residual Controllers and MPC

Residual learning augments a model-based controller with a learned correction: \(u = u_{\text{model-based}} + f_\theta(s)\). The neural network learns only the modelling error, making training more data-efficient and the combined controller safer.

Model Predictive Control (MPC) optimises a sequence of actions over a finite horizon \(H\):

$$\min_{u_{0:H-1}} \sum_{t=0}^{H-1} \ell(x_t, u_t) + \ell_f(x_H) \quad \text{s.t.} \quad x_{t+1} = f(x_t, u_t)$$

Only the first action is executed; the optimisation repeats at the next timestep (receding horizon). MPC naturally handles constraints (joint limits, contact forces) and provides anticipatory behaviour. MPPI (Model Predictive Path Integral) solves this via sampling: thousands of random trajectories are simulated, weighted by their costs, and combined via importance sampling — amenable to GPU parallelisation.

Key Insight: The trend in robot control is hybrid: model-based controllers provide stability and safety guarantees, while learned components handle unmodelled dynamics and task-specific adaptation. This "model-based + learned residual" paradigm combines the best of both worlds — interpretability and data efficiency.

References

  1. Hogan, N. (1985). Impedance control: An approach to manipulation. ASME Journal of Dynamic Systems, Measurement and Control, 107(1), 1–24.
  2. De Luca, A., & Oriolo, G. (1995). Modelling and control of nonholonomic mechanical systems. In Kinematics and Dynamics of Multi-Body Systems. Springer.
  3. Wensing, P. M., & Orin, D. E. (2013). Generation of dynamic humanoid behaviors through task-space control with conic optimization. ICRA.

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