Dynamics, actuators and control

1.1 Kinematics is not enough

Kinematics ignores mass: it says “joint 2 must be at 40° at time 0.5 s”. But a motor does not produce an angle, it produces a torque. To know which torque is needed we need dynamics: the relation between torques and motions. Without it, a robot cannot be sized (which motor?), simulated, or controlled well at high speed.

1.2 One link, by hand

A link of mass m whose centre of mass is at distance l from the joint, turning in a vertical plane (θ is measured from the horizontal). Newton for rotation: the torque of the motor, minus the torque of gravity, minus friction, equals inertia times angular acceleration.

τ = I θ̈ + b θ̇ + m g l cos θ

1.3 The equation of a robot

For a robot with n joints the same idea gives the dynamic model, found with the Lagrange method (energies) or the Newton–Euler method (forces, link by link):

M(q) q̈ + C(q, q̇) q̇ + g(q) + f(q̇) = τ

1.4 Using the model

• Sizing: find the peak and RMS torques of a task and choose motors and gearboxes (lesson 2).
• Simulation: integrate the equation to test a controller before touching the robot.
• Control: compute feedforward torques (lesson 5).
• Identification: the masses, centres of mass and friction are rarely known exactly; they are measured by running test motions and fitting the equation (it is linear in the inertial parameters).
• Payload check: a heavier tool changes M and g; the controller must be told the payload data.

Key terms

• dynamics
• torque
• inertia
• Lagrange / Newton–Euler
• Coriolis
• gravity compensation
• friction
• payload

Dynamics in the Master-level course →

2.1 The kinds of actuators

2.2 Why a gearbox?

An electric motor is fast and weak (for example 3000 rpm and 0.1 N·m); a robot joint is slow and strong (for example 30 rpm and 10 N·m). A gearbox with ratio N turns one into the other:

τ_joint = N · η · τ_motor, ω_joint = ω_motor / N, J_reflected = N² · J_motor

The efficiency η (0.6 to 0.95) accounts for the losses. The last relation matters a lot: the motor inertia seen from the joint is multiplied by N². With a high ratio, the motor inertia dominates the arm inertia, and the joints are almost independent of each other (lesson 5), but the robot is hard to back-drive (to push by hand) and forces cannot be felt through the gearbox.

Key terms

• brushless motor
• stepper
• gear ratio
• harmonic drive
• backlash
• reflected inertia
• RMS torque
• brake

Sensors that measure the robot itself are proprioceptive (like your sense of where your limbs are with your eyes closed). Those that measure the world are exteroceptive (cameras, lidar…; see the “Perception” course).

3.1 Resolution, accuracy, repeatability

• Resolution: the smallest change that the sensor can show. A 17-bit encoder gives 2¹⁷ = 131 072 counts per turn, i.e. 0.0027° per count.
• Accuracy: how close the reading is to the true value. A good encoder can have a high resolution but a poor accuracy (eccentric disc, temperature).
• Repeatability: how close the readings are when the same situation is repeated. Industrial robots are very repeatable (±0.02 mm) but less accurate (a fraction of a millimetre, until calibrated).
• Noise and quantisation: differentiating an encoder to get the speed amplifies both; speed is filtered (or estimated with an observer).

Key terms

• proprioceptive
• encoder
• homing
• resolution
• accuracy
• repeatability
• force / torque sensor
• IMU

4.1 Open loop and closed loop

Open loop: send a command and hope (a stepper motor, an oven timer). Closed loop (feedback): measure the result, compute the error e = wanted − measured, and correct. Feedback fights the things that the model does not know: friction, payload changes, a push. Almost all robot joints are closed loops.

4.2 The PID law

u(t) = K_p e(t) + K_i ∫ e(t) dt + K_d ė(t)

4.3 Tuning and good practice

• Manual tuning: start with K_i = K_d = 0, raise K_p until it is fast but starts to overshoot, add K_d to damp it, add a little K_i to remove the remaining error.
• Ziegler–Nichols (closed loop): with P only, raise K_p until the output oscillates steadily: you get the ultimate gain K_u and the period T_u. Then K_p = 0.6 K_u, T_i = T_u/2, T_d = T_u/8, i.e. K_i = K_p/T_i and K_d = K_p T_d. For K_u = 20 and T_u = 0.2 s: K_p = 12, K_i = 120, K_d = 0.3. It is a starting point, usually a bit aggressive.
• Anti-windup: when the actuator saturates (maximum current), stop integrating, otherwise the integral grows and causes a big overshoot afterwards. Try the saturation switch in the widget.
• Derivative on the measurement (not on the error) avoids a spike at each change of target; filter it to limit noise.
• Feedforward: add to the output the torque that the model predicts (speed, acceleration, friction, gravity). The feedback then only has small errors to correct.

4.4 Three nested loops

A servo drive does not use a single loop but three, from the fastest to the slowest: the current loop (torque, 10 to 20 kHz), the velocity loop (1 to 4 kHz), the position loop (0.5 to 1 kHz). Each loop is about 5 to 10 times slower than the one inside it, so that the inner loop looks “instant” to the outer one. Tune from the inside to the outside.

Key terms

• feedback
• error
• PID
• overshoot
• steady-state error
• windup
• cascaded loops
• feedforward

5.1 Independent joint control

The simplest controller gives each joint its own PID and ignores the others. In the dynamic model the coupling between joints is in M(q) and C. With a large gear ratio the motor inertia (N² J_m) is much bigger than the variation of the arm inertia, so each joint almost behaves like an isolated one and the other joints are small disturbances for the feedback. This is how most industrial robots are controlled at moderate speed.

5.2 PD plus gravity compensation

τ = K_p (q_d − q) − K_d q̇ + g(q)

The model is used only for the gravity term. This removes the steady-state error without any integral, and it can be proved that the robot converges to any target (Takegaki and Arimoto). The price: g(q) must be accurate (payload data!). This is the standard way to hold a robot still and to make a cobot light to hand-guide.

5.3 Computed torque (inverse dynamics control)

To follow a fast trajectory (q_d, q̇_d, q̈_d) the model is used completely:

τ = M(q) [ q̈_d + K_d (q̇_d − q̇) + K_p (q_d − q) ] + C(q, q̇) q̇ + g(q)

Put this torque in the robot equation: the terms C, g cancel exactly and M multiplies the bracket on both sides, leaving a linear error equation, the same for every joint and every pose:

ë + K_d ė + K_p e = 0

5.4 Beyond: robust, adaptive, optimal, learning

Key terms

• independent joint control
• gravity compensation
• computed torque
• inverse dynamics
• adaptive control
• MPC
• operational space

6.1 The problem of contact

Position control is a very stiff spring pulling towards the target. When the tool touches a rigid surface, a tiny position error becomes a huge force: with a surface of stiffness k_e = 10⁶ N/m (steel), an error of only 0.1 mm gives F = 10⁶ × 0.0001 = 100 N. Polishing, inserting a part, screwing, working next to humans: all need to control the force, not just the position.

6.2 Passive and active compliance

• Passive: soft pads, springs, a remote-centre-compliance (RCC) device for peg-in-hole, series elastic actuators. Fast, simple, but fixed.
• Active: the controller makes the robot behave as if it had springs, by measuring forces (wrist sensor, joint torque sensors, or motor current) and modifying the motion. Flexible and tunable.

6.3 Impedance and admittance

Impedance control (Hogan) makes the tool behave like a virtual mass-spring-damper around the target x_d:

M_d ë + D_d ė + K_d e = F_ext, e = x − x_d

With K_d = 500 N/m, a push that moves the tool by 10 mm costs 5 N: soft and safe. With K_d = 50 000 N/m the same 10 mm costs 500 N: stiff, precise. You choose the stiffness in each direction by software.

6.4 Hybrid position / force control

A task often has directions of motion and directions of contact. Wiping a table: force control along the normal (press with 10 N) and position control in the plane (follow the path). A selection matrix S (diagonal with 0 or 1) chooses, direction by direction, which law applies (Raibert and Craig). The force loop is slow and has a low gain, because with a stiff surface a fast force loop easily becomes unstable.

Key terms

• compliance
• stiffness
• impedance control
• admittance control
• hybrid position/force control
• hand guiding
• RCC

The speed laws (trapezoid) were seen in the other courses. Here is the practical side: what a robot programmer really chooses.

7.1 Corners: blending (zones)

Stopping exactly at every programmed point wastes time. With a blending radius (called zone, approximation, CNT or smoothing depending on the brand) the robot starts the next segment before reaching the corner and cuts it with a round curve. Large zone = faster and smoother, but the tool does not touch the point. “Fine / exact stop” means no blending.

7.2 Polynomial profiles

To go from q₀ to q_f in a time T with zero speed at both ends, with τ = t/T and Δq = q_f − q₀:

7.3 Time-optimal and orientation

• Time-optimal trajectory along a given path (TOPP): go as fast as the torque and speed limits allow at every point, using the dynamic model. Cycle-time winners in packaging use it.
• Orientation: interpolate with quaternions (SLERP), never with Euler angles (lesson 6 of the course on singularities).
• Synchronisation: in PTP the slowest joint decides the duration and the other joints are slowed to arrive together.
• Speed override: the operator scales the speed by 10 %, 50 %, 100 %: the whole profile is stretched in time.

Key terms

• PTP
• linear / circular move
• blending
• spline
• quintic
• S-curve
• jerk
• time-optimal

Speed laws in the Master-level course →

8.1 Servo drive and fieldbus

• Servo drive: converts the command to the three-phase current of a brushless motor (FOC, field-oriented control), reads the encoder, runs the current, speed and often position loops, and protects the motor (over-current, over-temperature).
• Command modes: torque (current), velocity or position mode. A research robot or a cobot often uses torque mode; an industrial robot uses position mode with the controller closing the outer loop.
• Fieldbus: the digital link between the controller and the drives: EtherCAT (very common, cycles of 250 µs to 1 ms, all axes synchronised), CANopen, PROFINET, Ethernet/IP. It replaces dozens of wires.
• Safety layer: a separate, certified circuit (safe torque off, safe limited speed, emergency stop) that does not rely on the normal software.

Key terms

• interpolator
• servo drive
• FOC
• fieldbus
• EtherCAT
• hard real time
• safe torque off

9.1 A first look at vibration

Model a joint with a flexible gearbox as two masses linked by a spring: the motor mass J_m and the load mass J_l, with a stiffness k. It resonates at:

f = (1 / 2π) √( k (1/J_m + 1/J_l) )

For k = 10 000 N·m/rad, J_m = 0.2 and J_l = 0.5 kg·m² (both seen at the joint): f = (1/2π) √(10 000 × (5 + 2)) = (1/2π) √70 000 ≈ 42 Hz. The position loop bandwidth must stay well under this frequency, or the controller will excite the resonance. A trajectory whose acceleration changes abruptly every 24 ms (about 42 Hz) would excite it: this is what a jerk-limited profile avoids.

9.2 Calibration of the dynamic parameters

To use the model of lesson 1 and 5, run an identification: move every joint with a rich, safe excitation trajectory, record positions and motor currents, and fit the parameters (masses, centres of mass, inertias, friction) by least squares. Also identify the payload: most controllers have a “load identification” routine. A wrong payload setting is a classic cause of poor tracking, protective stops on cobots, and shorter gearbox life.

Key terms

• stick-slip
• backlash
• flexibility
• resonance
• notch filter
• input shaping
• identification
• duty cycle