Robot Control Theory

Overview

The goal of robot control is to make the robot precisely execute desired motions or apply desired forces. The controller computes driving torques based on the dynamics model to achieve accurate tracking of position, velocity, and force.

graph TB
    subgraph "Control Architecture"
        A["Trajectory Planner"] --> B["Controller"]
        B --> C["Robot"]
        C -->|"Joint Encoders"| D["State Estimation"]
        C -->|"Force/Torque Sensors"| E["Force Feedback"]
        D --> B
        E --> B
    end
    style A fill:#e8f4fd,stroke:#2196F3
    style B fill:#fff3e0,stroke:#FF9800
    style C fill:#f3e5f5,stroke:#9C27B0

1 PID Control

1.1 Basic Form

PID is the most fundamental and widely used control method. For a single joint:

\[ u(t) = K_p e(t) + K_i \int_0^t e(\tau) \, d\tau + K_d \dot{e}(t) \]

where the error \(e(t) = q_d(t) - q(t)\).

Role of each term:

Term	Effect	Tuning Impact
P (Proportional)	\(K_p e\)	Reduces steady-state error; too large causes oscillation
I (Integral)	\(K_i \int e \, dt\)	Eliminates steady-state error; too large causes overshoot and oscillation
D (Derivative)	\(K_d \dot{e}\)	Adds damping, suppresses oscillation; sensitive to noise

1.2 Independent Joint PID

Treating each joint as an independent system, designing PID separately:

\[ \tau_i = K_{p,i} e_i + K_{d,i} \dot{e}_i + K_{i,i} \int e_i \, dt \]

Limitations:

Ignores dynamic coupling between joints
Ignores gravity compensation
Tracking accuracy degrades at high speeds

1.3 PD + Gravity Compensation

One of the most commonly used simple control laws in practice:

\[ \tau = K_p e + K_d \dot{e} + g(q) \]

where \(g(q)\) is the gravity compensation term. This control law ensures accuracy under static and low-speed conditions.

Asymptotic Stability Proof

Choosing the Lyapunov function \(V = \frac{1}{2}e^T K_p e + \frac{1}{2}\dot{q}^T M(q)\dot{q}\), one can prove that the system is asymptotically stable under PD + gravity compensation.

2 Computed Torque Control

2.1 Basic Idea

Utilize the complete dynamics model for nonlinear feedback linearization, transforming the nonlinear system into a linear one.

Control Law:

\[ \tau = M(q)\ddot{q}_d + C(q, \dot{q})\dot{q} + g(q) + K_p e + K_d \dot{e} \]

More compactly written as:

\[ \tau = M(q)[\ddot{q}_d + K_p e + K_d \dot{e}] + C(q, \dot{q})\dot{q} + g(q) \]

2.2 Closed-Loop Analysis

Substituting the control law into the equations of motion \(M\ddot{q} + C\dot{q} + g = \tau\):

\[ M(q)[\ddot{q}_d + K_p e + K_d \dot{e} - \ddot{q}] = 0 \]

Since \(M(q)\) is positive definite, the error dynamics become:

\[ \ddot{e} + K_d \dot{e} + K_p e = 0 \]

This is a linear second-order system! Choosing \(K_p, K_d > 0\) guarantees exponential stability.

2.3 Practical Considerations

Issue	Description	Solution
Model Uncertainty	\(M, C, g\) deviate from true values	Robust control, adaptive control
Computational Delay	Real-time inverse dynamics computation	Newton-Euler \(O(n)\) recursion
Joint Friction	Unmodeled friction forces	Friction compensation terms
Sensor Noise	Inaccurate velocity estimates	Filtering, observers

3 Impedance Control

3.1 Motivation

During environmental interaction (e.g., assembly, wiping), pure position control is too "rigid." Impedance control adjusts the dynamic behavior of the end-effector to achieve compliant interaction.

3.2 Target Impedance

The desired end-effector dynamic behavior:

\[ M_d \ddot{e} + D_d \dot{e} + K_d e = F_{\text{ext}} \]

where:

Parameter	Meaning	Effect
\(M_d\)	Desired inertia	Controls acceleration response
\(D_d\)	Desired damping	Controls velocity response
\(K_d\)	Desired stiffness	Controls displacement response
\(e = x_d - x\)	End-effector position error
\(F_{\text{ext}}\)	External contact force

3.3 Implementation

Joint-Space Impedance Control:

\[ \tau = M(q)\ddot{q}_r + C(q,\dot{q})\dot{q}_r + g(q) - J^T(q) F_{\text{ext}} \]

where the reference acceleration \(\ddot{q}_r\) is determined by the impedance relationship.

Simplified Implementation (without force sensor):

\[ \tau = J^T(q)[K_d(x_d - x) + D_d(\dot{x}_d - \dot{x})] + g(q) \]

Here the impedance is determined by \(K_d\) and \(D_d\), requiring no force sensor.

3.4 Admittance Control

The dual form of impedance control:

Impedance Control: Input motion deviation, output force \(\rightarrow\) suitable for force-controlled actuators
Admittance Control: Input force deviation, output motion \(\rightarrow\) suitable for position-controlled actuators

\[ x_c = M_d^{-1}(F_{\text{ext}} - D_d \dot{x}_c - K_d x_c) \]

4 Force Control

4.1 Hybrid Force/Position Control

Raibert-Craig method (1981): Control force and position separately in different directions.

Define a selection matrix \(S\) (diagonal, elements are 0 or 1):

\[ \tau = J^T [S \cdot F_{\text{force\_ctrl}} + (I - S) \cdot F_{\text{pos\_ctrl}}] \]

\(S_{ii} = 1\): Force control in the \(i\)-th direction
\(S_{ii} = 0\): Position control in the \(i\)-th direction

Example: Force control perpendicular to the workpiece surface, position control parallel to it.

4.2 Force Controller Design

PI Force Control:

\[ F_{\text{force\_ctrl}} = K_{fp}(F_d - F_{\text{meas}}) + K_{fi} \int (F_d - F_{\text{meas}}) \, dt \]

Challenges of Force Control

Discontinuity at contact/non-contact state transitions
Stability is difficult to guarantee with unknown environment stiffness
Force sensor noise and delays

5 Operational Space Control

5.1 Basic Idea

Design control laws directly in task space (Cartesian space) and map to joint space via the Jacobian.

Operational Space Dynamics (Khatib, 1987):

\[ \Lambda(x) \ddot{x} + \mu(x, \dot{x}) + p(x) = F \]

where:

\[ \Lambda(x) = (J M^{-1} J^T)^{-1} \]

\[ \mu = \Lambda J M^{-1} C \dot{q} - \Lambda \dot{J} \dot{q} \]

\[ p = \Lambda J M^{-1} g \]

5.2 Control Law

\[ F = \Lambda(x)[\ddot{x}_d + K_p e + K_d \dot{e}] + \mu(x, \dot{x}) + p(x) \]

Mapped to joint torques:

\[ \tau = J^T F \]

Equivalently:

\[ F = J^{-T} \tau \]

5.3 Null-Space Control for Redundant Robots

For redundant robots (\(n > 6\)), secondary tasks can be accomplished in the null space while satisfying the primary task:

\[ \tau = J^T F_{\text{task}} + (I - J^T J^{-T}) \tau_0 \]

where \(\tau_0\) is the null-space torque, which can be used for:

Joint limit avoidance
Singularity avoidance
Self-collision avoidance
Posture optimization

6 Adaptive Control

6.1 Motivation

When dynamics parameters (mass, inertia, friction coefficients, etc.) are unknown or changing, online estimation and compensation are needed.

6.2 Adaptive Computed Torque

Leveraging the linear parameterization property of the dynamics equations:

\[ M(q)\ddot{q} + C(q, \dot{q})\dot{q} + g(q) = Y(q, \dot{q}, \ddot{q}) \, \hat{\pi} \]

Control law:

\[ \tau = Y(q, \dot{q}, \dot{q}_r, \ddot{q}_r) \hat{\pi} + K_d s \]

where \(s = \dot{e} + \Lambda e\) is the sliding variable and \(\hat{\pi}\) is the parameter estimate.

Parameter Update Law:

\[ \dot{\hat{\pi}} = -\Gamma Y^T s \]

where \(\Gamma > 0\) is the learning rate matrix.

Stability Guarantee

Through Lyapunov analysis, it can be proven that \(s \to 0\) (tracking error converges), but \(\hat{\pi} \to \pi\) is not guaranteed (parameters do not necessarily converge to true values unless the persistent excitation condition is satisfied).

7 Robust Control

7.1 Sliding Mode Control

Design a sliding surface \(s = \dot{e} + \lambda e\); the control law ensures \(s\) reaches zero quickly and stays there:

\[ \tau = \hat{M}(q)[\ddot{q}_d + \lambda\dot{e}] + \hat{C}\dot{q} + \hat{g} - K \, \text{sgn}(s) \]

where \(K\) is large enough to overcome model uncertainty.

Chattering Problem

The sign function \(\text{sgn}(s)\) causes high-frequency chattering near \(s=0\). In practice, a saturation function \(\text{sat}(s/\phi)\) is used instead, where \(\phi\) is the boundary layer thickness.

7.2 \(H_\infty\) Control

Designed in the frequency domain, minimizing the \(\infty\)-norm of the transfer function from disturbance to performance output:

\[ \min_K \|T_{zw}(s)\|_\infty \]

Suitable for handling structured uncertainty and external disturbances.

8 Learning-Based Control

8.1 Iterative Learning Control (ILC)

For repetitive tasks, use the error from the previous execution to improve the next control input:

\[ u_{k+1}(t) = u_k(t) + L \cdot e_k(t) \]

8.2 Reinforcement Learning Control

Directly learn the control policy without relying on an accurate model:

Model-Free: PPO, SAC directly optimize the policy
Model-Based: Learn a dynamics model + MPC
Sim-to-Real: Train in simulation, transfer to the real robot

8.3 Real-Time System Requirements

Robot control has strict real-time requirements (see Real-Time Systems):

Control Layer	Frequency	Latency Requirement
Joint Servo	1-10 kHz	< 1 ms
Force Control	500 Hz - 1 kHz	< 2 ms
Motion Control	100-500 Hz	< 10 ms
Planning Layer	1-50 Hz	< 100 ms

9 Control Architecture Summary

graph TB
    A["Task Planning"] --> B["Trajectory Generation"]
    B --> C{"Control Strategy Selection"}
    C -->|"Known Model"| D["Computed Torque Control"]
    C -->|"Model Uncertain"| E["Adaptive Control"]
    C -->|"Disturbances"| F["Robust Control"]
    C -->|"Interaction Needed"| G["Impedance/Force Control"]
    C -->|"No Model"| H["Learning-Based Control"]
    D --> I["Joint Servo"]
    E --> I
    F --> I
    G --> I
    H --> I
    I --> J["Robot"]
    style C fill:#fff3e0,stroke:#FF9800
    style I fill:#e8f4fd,stroke:#2196F3

10 Common Tools and Frameworks

Tool	Purpose
Simulink / MATLAB	Classical control design and simulation
Drake	Model predictive control, trajectory optimization
ros2_control	ROS2 control framework
MuJoCo	Dynamics simulation (including contacts)
IsaacGym / IsaacSim	GPU-parallel simulation + RL training

References

Siciliano, B. et al. (2009). Robotics: Modelling, Planning and Control. Springer.
Slotine, J.-J. E. & Li, W. (1991). Applied Nonlinear Control. Prentice-Hall.
Khatib, O. (1987). A unified approach for motion and force control of robot manipulators. IEEE RA.
Craig, J. J. (2005). Introduction to Robotics: Mechanics and Control. Pearson.
Spong, M. W. et al. (2006). Robot Modeling and Control. Wiley.