Robot Dynamics and Control

Introduction

Robot dynamics describes how forces and torques produce motion, while robot control determines how actuators should be commanded to obtain desired motion or interaction with the environment. Kinematics relates position, velocity, and acceleration without considering causes; dynamics adds mass, inertia, gravity, friction, actuator limitations, and external forces. A reliable dynamic model is central to simulation, actuator sizing, trajectory planning, feedback control, collision detection, and safe human–robot interaction.

For an n-joint serial manipulator, the general joint-space model is

Here, q is the n×1 joint-position vector in radians or metres, q̇ and q̈ are joint velocity and acceleration, M(q) is the positive-definite inertia matrix, C(q,q̇)q̇ represents Coriolis and centrifugal effects, g(q) is the gravity vector, f(q̇) represents friction, τ is the actuator-torque vector in N·m or actuator-force vector in N, J(q) is the Jacobian, and Fₑ is an external wrench in N and N·m. This equation assumes rigid links, known geometry and parameters, and a chosen generalized-coordinate description.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain the physical meaning of robot dynamic terms.
2. Formulate manipulator dynamics using Newton–Euler and Lagrangian methods.
3. Identify assumptions, coordinate conventions, and parameter requirements.
4. Analyse equilibrium and closed-loop stability using energy and state-space ideas.
5. Design joint, trajectory, force, impedance, and hybrid controllers.
6. Compare PID, computed-torque, state-space, adaptive, and robust methods.
7. Implement a controller while accounting for sampling, saturation, friction, sensing, and safety.

1. Foundations of Robot Dynamics

A manipulator has generalized coordinates q, link masses mᵢ, centres of mass, inertia tensors Iᵢ, joint axes, and actuator characteristics. The kinetic energy is the sum of translational and rotational energies:

where vᵢ and ωᵢ are the centre-of-mass linear and angular velocities, and Iᵢ is expressed in a consistent frame. The potential energy due to gravity is

where g₀ is the gravitational acceleration vector and rᵢ is the centre-of-mass position. Depending on the chosen vertical-axis convention, the sign of g₀ changes; the physically important requirement is consistency.

The Lagrangian is

For ideal conservative links, the Euler–Lagrange equation is

For all coordinates, this yields a vector equation. Non-conservative effects such as viscous friction can be included by adding a generalized force Qᵢ, giving the right-hand side τᵢ + Qᵢ. Coulomb friction is often approximated by F_c sign(q̇ᵢ), although this approximation is discontinuous at zero velocity and may require a smooth model in numerical simulation.

Important structural properties are:

• M(q) is symmetric and positive definite for physically valid configurations.
• C(q,q̇)q̇ contains velocity-coupling effects.
• The gravity vector can be obtained from potential energy: g(q) = ∂V/∂q.
• With a suitable definition of C, Ṁ − 2C is skew-symmetric. This property is fundamental in Lyapunov proofs.
• Robot dynamics are nonlinear, coupled, and configuration dependent.

2. Lagrangian Formulation

The Lagrangian method is systematic and especially convenient for deriving joint-space equations. A practical procedure is:

1. Select generalized coordinates and positive joint directions.
2. Determine each link’s centre-of-mass velocity and angular velocity.
3. Calculate T and V.
4. Form L = T − V.
5. Apply the Euler–Lagrange equation.
6. Add friction, actuator dynamics, and external generalized forces.

Worked example: a single rotary link

Consider a rigid link of length l, mass m, with its centre of mass at l_c from a revolute joint. Let q be measured from the upward vertical, and let the link have mass moment of inertia I about the joint axis. Its kinetic and potential energies are

Thus L = T − V. Applying the Euler–Lagrange equation gives

If q is measured from the downward vertical instead, the gravity term has the opposite sign. This illustrates why angle conventions must be documented. Adding viscous friction bq̇ gives

The units are consistent: inertia times angular acceleration is N·m, bq̇ is N·m when b has units N·m·s/rad, and mgl_c is N·m.

3. Newton–Euler Formulation

The Newton–Euler method applies force and moment balances to every link. It is computationally efficient for recursive implementation and is widely used in real-time inverse dynamics.

For a rigid body with mass m, centre-of-mass acceleration a_c, angular velocity ω, angular acceleration α, and inertia tensor I about the centre of mass:

F is the resultant force and N_c is the resultant moment about the centre of mass. The cross product term is the gyroscopic contribution. In a serial chain, a forward recursion computes link velocities and accelerations from the base outward. A backward recursion computes forces and moments from the end effector toward the base, then projects the joint-axis moment or force to obtain τ.

The Newton–Euler formulation is attractive because it avoids explicitly constructing every symbolic term in M, C, and g. Its disadvantages are sensitivity to frame conventions and the need for careful treatment of spatial transforms, joint offsets, and external wrenches.

4. State-Space Representation and Stability

Define the state x = [qᵀ q̇ᵀ]ᵀ. If M(q) is invertible, the robot can be written as

For an equilibrium (qₑ, 0) with no external wrench, the required holding torque is τₑ = g(qₑ) + f(0), subject to the friction model. Linearising around an equilibrium gives

where A and B are Jacobian matrices of the nonlinear state equation. A continuous-time linear system is locally asymptotically stable if every eigenvalue of A has a strictly negative real part. This conclusion applies to the linearised model and therefore establishes local, not automatically global, stability of the nonlinear robot.

For nonlinear robot control, a common Lyapunov candidate is the tracking energy

where e = q − q_d is position error, s = ė + Λe is a filtered error, Kₚ and Λ are positive-definite matrices. A controller designed so that

with Kᵥ positive definite makes the error dynamics stable under ideal model and measurement assumptions. If V̇ is only non-positive, additional arguments such as LaSalle’s invariance principle may be needed to establish convergence.

5. Joint and Trajectory Control

PID control

For a single joint, the basic PID law is

where e = q_d − q. Kₚ, Kᵢ, and K_d have units selected so each term is torque. In digital control, the integral is approximated by a sum and the derivative by a difference. Derivative action should generally use measured velocity or a filtered error because numerical differentiation amplifies noise.

PID is simple and effective when gravity and coupling are modest or compensated separately. Practical implementation requires anti-windup: when actuator saturation prevents the requested torque, the integral state must be limited or back-calculated. Otherwise, recovery after saturation can be slow and can cause overshoot.

Gravity compensation

A useful improvement is

which removes the steady torque needed to hold a static posture, assuming the gravity model is accurate. It does not compensate for inertial coupling during rapid motion.

Computed-torque control

Let e = q − q_d and choose s = ė + Λe. A model-based control law is

where f̂ is an estimated friction term. Under exact modelling and no disturbances, the resulting error equation is approximately

and e converges when Λ and Kᵥ are positive definite. In practice, parameter errors, unmodelled flexibility, sensor noise, and delay prevent perfect cancellation. High gains can improve disturbance rejection but may excite structural resonances.

Feedforward and feedback

Trajectory control commonly combines inverse-dynamics feedforward with feedback. Feedforward anticipates the torque required by q_d, q̇_d, and q̈_d; feedback corrects modelling errors and disturbances. Feedforward alone is not robust because it cannot correct errors caused by friction, payload changes, or external contact.

6. Force, Impedance, and Hybrid Control

Position control is appropriate for free-space motion, but contact tasks require regulation of interaction forces as well as position.

Force control

A force controller compares desired and measured contact wrench:

The commanded wrench is transformed into joint torque by τ = JᵀF_c, subject to the Jacobian convention and wrench-frame definition. Force sensing may come from a wrist force–torque sensor, joint-torque estimation, or motor-current measurement. Contact dynamics, sensor noise, friction, and compliance strongly affect stability.

Impedance control

Impedance control specifies a desired relationship between motion error and interaction force:

Here x_e is task-space displacement error, M_d, D_d, and K_d are desired inertia, damping, and stiffness matrices, and F_e is measured environmental force. A compliant robot may move under contact force rather than attempting to maintain exact position. This is often safer for assembly and human interaction.

Admittance control is conceptually complementary: measured force is converted into a motion command through a virtual mass–damper–spring model, and an inner position loop tracks that command. Impedance is often easier when torque control is available; admittance is often practical for position-controlled industrial robots with an external force sensor.

Hybrid position–force control

Hybrid control assigns position regulation to selected task directions and force regulation to complementary directions. For example, during surface polishing, motion tangential to the surface may be position-controlled while normal force is regulated. A selection matrix S identifies force-controlled directions:

The matrix I is the identity matrix. This simple expression represents a conceptual decomposition; actual implementation must account for the contact geometry, Jacobian rank, coordinate transformations, and coupled dynamics.

7. Adaptive and Robust Control

Adaptive control

Robot dynamics can be expressed in a linearly parameterised form:

where Y is a known regressor matrix and θ contains unknown but constant physical parameters such as masses, centres of mass, and inertial coefficients. An adaptive controller estimates θ online. A typical parameter update has the form

where Γ is a positive-definite adaptation-gain matrix. Adaptation can compensate for payload changes, but parameter convergence is not guaranteed unless the motion provides sufficient excitation. Safety limits are essential because estimates can drift under sensor bias or unmodelled disturbances.

Robust control

Robust methods explicitly tolerate bounded uncertainty. If the model error is bounded by d̄, a robust term may be designed to dominate it. Sliding-mode control, for example, uses a discontinuous or smoothed term related to sign(s). It can provide strong disturbance rejection but may generate chattering, which excites unmodelled dynamics and causes actuator wear. Boundary layers, higher-order sliding modes, or continuous robust approximations reduce chattering at the cost of finite tracking accuracy.

Other approaches include H∞ control, disturbance observers, passivity-based control, and model predictive control. The appropriate method depends on computation, uncertainty, actuator bandwidth, certification requirements, and the consequences of failure.

8. Implementation Architecture

A practical robot controller typically contains the following layers:

1. Task and trajectory layer: generates q_d, q̇_d, q̈_d or task-space references.
2. Control layer: computes torque, velocity, or position commands.
3. Low-level actuator layer: closes current, velocity, and motor-position loops.
4. Estimation layer: filters position, estimates velocity, payload, and external torque.
5. Safety layer: enforces limits, watchdogs, collision reactions, and emergency stopping.

Sampling frequency must be sufficiently higher than the closed-loop bandwidth. Time stamps, deterministic communication, and synchronized sensing are important because delay introduces phase lag and can destabilize high-gain loops. Quantization, encoder resolution, motor dead zones, backlash, cable elasticity, and thermal limits should be included in testing.

A sound commissioning sequence is: validate frames and signs; test one joint at low speed; verify gravity compensation; identify inertial and friction parameters; test saturation and emergency stop; increase gains gradually; then validate coordinated motion and contact tasks. Simulation is valuable, but hardware-in-the-loop and controlled physical tests are required because contact and actuator nonlinearities are difficult to model completely.

9. Illustrative Case: Robotic Peg Insertion

Background

Consider a six-axis industrial manipulator inserting a metal peg into a machined hole. The task requires rapid free-space positioning followed by controlled contact. The case is illustrative rather than a report of a particular commercial system.

Problem

Pure position control may cause high lateral forces when the peg is slightly misaligned. Excess force can damage the part, stall the motor, or create a safety hazard.

Analysis

The robot uses a dynamic model for gravity compensation and trajectory feedforward. A wrist force–torque sensor measures contact forces. The insertion axis is selected for force control, while lateral directions use compliant impedance. The desired normal force is F_d = 30 N. A force error of 5 N with proportional gain 0.4 (in appropriate controller units) produces a corrective command of 2 controller units; the exact resulting torque depends on the Jacobian and the chosen inner-loop scaling.

Approach

During approach, computed-torque position control is used. At contact detection, the controller smoothly switches to an impedance/force mode. The normal axis regulates force, and tangential axes use low virtual stiffness so that small alignment errors produce lateral motion rather than excessive force. Torque and force limits override task commands. A filtered force signal prevents sensor noise from causing abrupt commands.

Outcome

The hybrid strategy reduces peak contact force and allows the peg to self-align within the mechanical clearance. Performance should be assessed using insertion time, peak force, force variance, failure rate, and part damage—not merely position error.

Learning points

The case demonstrates that the “best” controller depends on task phase. Model-based position control is efficient in free space, whereas compliant force interaction is more appropriate during contact. Mode transitions, sensor calibration, saturation handling, and safety supervision are as important as the nominal control law.

10. Limitations, Safety, and Contemporary Practice

No dynamic model is exact. Flexible links, gearbox compliance, temperature-dependent friction, payload uncertainty, cable forces, structural vibration, and contact deformation produce discrepancies. Identification should therefore report parameter uncertainty and validation error rather than presenting one fitted model as universally correct.

Robot control also has ethical and regulatory dimensions. A controller must respect workspace restrictions, speed and force limits, safeguarding requirements, risk assessment, and applicable machinery and collaborative-robot standards. Human–robot systems should fail safely when sensing or communication is lost. Logs, access control, software versioning, and change approval support accountability.

Contemporary developments include torque-controlled collaborative robots, learning-based residual dynamics, digital twins, reinforcement learning for policy adaptation, and model predictive control. Machine learning can improve friction or payload estimation, but a learned component should be bounded by a verified safety controller. Training data may not cover unusual contacts, and opaque policies are unsuitable as the sole protection against hazardous motion. The strongest engineering practice combines analytical dynamics, measured data, conservative constraints, and runtime monitoring.

Chapter Summary

Robot dynamics connects joint motion with actuator torque, gravity, inertia, velocity coupling, friction, and external forces. Lagrangian modelling derives equations from kinetic and potential energy, while Newton–Euler modelling uses recursive force and moment balances and is efficient for real-time computation. The standard manipulator equation uses the inertia matrix M, Coriolis/centrifugal term Cq̇, gravity g, friction, actuator torque, and Jacobian-transformed external wrench.

State-space models support linearisation, eigenvalue analysis, observers, and modern control design. Lyapunov methods use the robot’s energy structure to establish stability under suitable positive-definiteness and modelling assumptions. PID and gravity compensation are practical baseline methods; computed torque adds model-based cancellation and trajectory feedforward. Force, impedance, admittance, and hybrid controllers address contact tasks. Adaptive control estimates uncertain parameters, while robust control tolerates bounded modelling errors.

Successful implementation requires correct frames, calibrated sensors, realistic friction and payload models, sampling and delay analysis, saturation and anti-windup protection, staged commissioning, and independent safety supervision. A controller must be evaluated not only by tracking error but also by force peaks, energy, robustness, failure rate, and safe behaviour under uncertainty.

Key Terms

Dynamic model: Mathematical relationship between motion, forces, and torques.

Generalized coordinate: Coordinate describing an independent degree of freedom.

Inertia matrix: Configuration-dependent matrix relating joint acceleration to inertial torque.

Coriolis effect: Velocity coupling caused by motion of multiple coordinates.

Centrifugal effect: Velocity-squared coupling associated with rotating coordinates.

Gravity vector: Joint torque required to balance gravitational loading.

Jacobian: Matrix relating joint velocities to task-space velocities.

Wrench: Combined force and moment vector.

Lagrangian: Kinetic energy minus potential energy.

Newton–Euler method: Force and moment balance method for rigid-body dynamics.

PID control: Proportional, integral, and derivative feedback control.

Computed-torque control: Model-based nonlinear feedback linearisation method.

Trajectory tracking: Following desired position, velocity, and acceleration histories.

Impedance control: Regulation of the relationship between motion and interaction force.

Admittance control: Conversion of measured force into a motion command.

Hybrid control: Coordinated position and force control in different directions.

Adaptive control: Control that updates uncertain model parameters online.

Robust control: Control designed to tolerate bounded uncertainty or disturbance.

Lyapunov stability: Stability analysis based on a scalar energy-like function.

Anti-windup: Method preventing integral accumulation during actuator saturation.

Review Questions

Short Answer Questions

1. What physical effects are represented by M(q), C(q,q̇)q̇, and g(q) in the manipulator dynamic equation?
2. State two differences between Lagrangian and Newton–Euler dynamic formulations.
3. Why is gravity compensation useful in a joint controller?
4. Distinguish impedance control from admittance control.
5. Why is actuator saturation important in PID implementation?

Long Answer Questions

1. Derive the dynamics of a single rotary link using the Euler–Lagrange method and clearly state the angle convention.
2. Explain the forward and backward recursions used in the Newton–Euler method.
3. Describe computed-torque trajectory control and discuss the effects of model uncertainty.
4. Explain how Lyapunov analysis can establish stability for a filtered tracking error.
5. Compare PID, adaptive, and robust control for a robot with uncertain payload and friction.

Analytical / Application Questions

1. Design a control architecture for a robotic polishing task in which tangential position and normal force must both be regulated. Identify sensors, controlled directions, safety limits, and likely failure modes.
2. A two-joint robot exhibits oscillation after derivative gain is increased. Analyse possible causes involving sampling, filtering, structural resonance, delay, and actuator bandwidth, and propose a commissioning procedure.
3. Develop an experimental plan to identify gravity and friction parameters for a manipulator. Specify measured variables, excitation requirements, validation tests, and methods for reporting uncertainty.

Multiple Choice Questions with Answers

Question 1

Which term in the standard manipulator equation represents configuration-dependent resistance to joint acceleration?

A. M(q)

B. g(q)

C. JᵀFₑ

D. f(q̇)

Correct Answer: A. M(q)

Question 2

The Lagrangian of a mechanical system is defined as:

A. T + V

B. T − V

C. V − T

D. T ÷ V

Correct Answer: B. T − V

Question 3

Which method is generally most suitable for efficient recursive real-time inverse dynamics?

A. Newton–Euler

B. Static graphical analysis

C. Dimensional analysis alone

D. Open-loop interpolation

Correct Answer: A. Newton–Euler

Question 4

In a PID controller, anti-windup primarily addresses:

A. Encoder resolution

B. Integral accumulation during actuator saturation

C. Gravity modelling

D. Jacobian singularity only

Correct Answer: B. Integral accumulation during actuator saturation

Question 5

For ideal computed-torque control, positive-definite feedback gains are used to obtain:

A. Unbounded tracking error

B. Stable filtered-error dynamics

C. Zero actuator bandwidth

D. A singular inertia matrix

Correct Answer: B. Stable filtered-error dynamics

Question 6

Impedance control specifies a desired relationship between:

A. Network bandwidth and packet size

B. Motion error and interaction force

C. Mass and temperature only

D. Encoder counts and pixels

Correct Answer: B. Motion error and interaction force

Question 7

In hybrid position–force control, a selection matrix is used to:

A. Choose directions assigned to force or position regulation

B. Calculate link mass automatically

C. Remove all sensor noise

D. Guarantee global stability

Correct Answer: A. Choose directions assigned to force or position regulation

Question 8

Adaptive control is particularly intended to:

A. Estimate uncertain model parameters online

B. Eliminate the need for feedback

C. Make all links massless

D. Prevent every possible collision without sensing

Correct Answer: A. Estimate uncertain model parameters online

Question 9

A continuous-time linearised system is locally asymptotically stable when:

A. All eigenvalues have positive real parts

B. At least one eigenvalue is zero

C. All eigenvalues have strictly negative real parts

D. The input matrix is rectangular

Correct Answer: C. All eigenvalues have strictly negative real parts

Question 10

Why should a learned dynamics component usually be combined with a safety controller?

A. Learning systems never use data

B. Training data may not cover unusual or hazardous conditions

C. Analytical models cannot represent gravity

D. Safety controllers increase uncertainty by definition

Correct Answer: B. Training data may not cover unusual or hazardous conditions