Robot Statics and Dynamics

Introduction

Robot motion is governed not only by geometry and velocity but also by force, torque, mass distribution, and energy. Statics determines the forces and joint torques required to hold a manipulator in equilibrium. Dynamics determines the torques required to produce a specified motion, including the effects of inertia, gravity, Coriolis and centrifugal forces, and friction.

These analyses support actuator selection, gearbox sizing, trajectory planning, control-system design, payload evaluation, structural verification, and safe operation. The central dynamic model of an n-joint rigid manipulator is

where q is the vector of joint coordinates, M(q) is the inertia matrix, C(q,q̇)q̇ represents velocity-dependent Coriolis and centrifugal effects, g(q) is the gravity vector, τ_f is joint friction, τ_ext represents generalized external loads, and τ is the actuator-torque vector. The equation assumes rigid links, known mass properties, and a chosen set of generalized coordinates.

Learning Objectives

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

1. Distinguish robot statics from robot dynamics.
2. Calculate forces, moments, payload reactions, and simple joint torques.
3. Formulate manipulator dynamics using Newton–Euler and Euler–Lagrange methods.
4. Explain inertia, Coriolis, centrifugal, gravity, and friction effects.
5. Interpret the standard manipulator equation for design and control.
6. Apply dynamic reasoning to actuator sizing and industrial robot operation.

1. Foundations of Robot Statics

A body is in static equilibrium when its linear and angular accelerations are zero. For a rigid body:

Here ∑F is the vector sum of external forces and ∑M_O is the vector sum of moments about point O. Forces are measured in newtons (N), moments in newton-metres (N·m), and distances in metres (m).

The moment of a force F applied at position vector r from O is

The magnitude is M = rF sin φ, where φ is the angle between r and F. A perpendicular force produces the largest moment, M = rF. A force whose line of action passes through O produces zero moment about O.

For a planar revolute joint, the actuator torque must balance the moments produced by gravity, payload forces, friction, and any externally applied force. In an ideal static model,

where τ_g is the gravity torque and τ_ext is the generalized torque caused by external loads. This expression neglects acceleration and friction; those terms must be added for real systems.

Center of Mass and Payload

For point masses m_i at positions r_i, the center of mass is

The total gravitational force is W = mg, where m is mass and g ≈ 9.81 m/s² near Earth’s surface. A payload changes both the total gravitational load and the location of the combined center of mass. Consequently, payload capacity is limited not only by motor torque but also by bearing loads, deflection, braking capability, and structural strength.

Jacobian and Static Force Transmission

Let x denote the end-effector position and q the joint-coordinate vector. Virtual displacement gives

where J(q) is the manipulator Jacobian. Equating virtual work at static equilibrium,

leads to

Here F is the Cartesian wrench expressed consistently with the Jacobian. For a full spatial wrench, F contains forces and moments, and J is the corresponding 6 × n geometric or analytical Jacobian. The formula assumes ideal joints and virtual work expressed in compatible coordinates. Near a singular configuration, some Cartesian loads may require very large joint torques.

2. Worked Static Example: Two-Link Planar Arm

Consider a horizontal two-link planar arm with revolute joints. Link lengths are l₁ = 0.40 m and l₂ = 0.30 m. A 5 kg payload is held at the end. The arm configuration is q₁ = 30° and q₂ = 45°, where q₂ is the relative angle of link 2. Ignore link masses and friction.

The payload position is

For a downward payload force F_y = −mg, the gravitational joint torques can be obtained from the Jacobian. The planar position Jacobian is

Since τ = JᵀF and F = [0, −mg]ᵀ, the magnitudes opposing gravity are

With g = 9.81 m/s², cos 30° = 0.866, and cos 75° = 0.259:

τ₁ = 5 × 9.81 × [0.40(0.866) + 0.30(0.259)] ≈ 19.5 N·m.

τ₂ = 5 × 9.81 × [0.30(0.259)] ≈ 3.82 N·m.

The first joint supports the payload moment about the base, whereas the second joint supports only the moment of the payload about joint 2. Link masses, gearbox efficiency, acceleration, and safety factors would increase the required motor rating.

3. Rigid-Body Dynamics

Dynamics relates motion to force and torque. Translational motion is governed by Newton’s second law:

where a is the acceleration of the center of mass. Rotational motion about the center of mass is

where I_C is the 3 × 3 inertia tensor, α is angular acceleration, and ω is angular velocity. The cross-product term accounts for three-dimensional rotational coupling. In a principal-axis frame with diagonal inertia tensor, the principal-axis equations are Euler’s equations:

with cyclic equivalents for M₂ and M₃. These equations assume a rigid body and an inertial reference frame for the force and moment balance.

Inertia

The scalar moment of inertia about an axis is

where r is the perpendicular distance from the mass element dm to the axis. Its unit is kg·m². A slender rod of mass m and length L has I = mL²/12 about an axis through its center and perpendicular to the rod, and I = mL²/3 about a parallel axis through one end.

The parallel-axis theorem is

where d is the distance between parallel axes through O and the center of mass C. Incorrect inertia values can produce large errors in predicted acceleration and actuator torque.

4. Manipulator Dynamics in Standard Form

For an n-degree-of-freedom robot, the joint-space dynamic model is commonly written as

The terms have the following meanings:

The inertia matrix M(q) is symmetric and positive definite for a physically valid, nondegenerate robot. It changes with configuration because the relative positions of masses change. Coriolis terms are proportional to products of joint velocities, such as q̇_iq̇_j. Centrifugal terms commonly appear as q̇_i². The distinction depends on the chosen representation; both are velocity-dependent effects.

Gravity torque depends on configuration but not velocity. It can be derived from potential energy V(q):

where the gradient is taken with respect to q. For prismatic joints, the corresponding generalized gravity term has units of force rather than torque.

Friction may include viscous and Coulomb components:

where B is a viscous-friction coefficient and τ_c is Coulomb friction. The sign function is discontinuous at zero velocity, so practical models often use a smooth approximation or include static friction and stiction separately. Friction coefficients depend on temperature, lubrication, wear, gearbox preload, and velocity.

5. Euler–Lagrange Formulation

The Euler–Lagrange method is energy-based. Define the Lagrangian

where T is total kinetic energy and V is gravitational potential energy. For each generalized coordinate q_i,

If nonconservative forces exist, they are included through generalized forces Q_i:

The method assumes appropriately defined generalized coordinates and an accurate energy model. It is particularly efficient for serial manipulators because the scalar energies can be assembled systematically.

For a planar two-link robot, the kinetic energy can be written generally as

and the potential energy as V(q). Substitution into the Euler–Lagrange equations yields the standard matrix equation. The derivation is often shorter than writing force balances for every link, but obtaining accurate mass, center-of-mass, and inertia parameters remains essential.

Properties Useful for Control

A valid robot model satisfies several structural properties:

• M(q) is symmetric and positive definite.
• The gravity vector is the gradient of potential energy.
• With a suitable definition of C, the matrix Ṁ − 2C is skew-symmetric.
• Therefore, q̇ᵀ[Ṁ − 2C]q̇ = 0.

The last property supports energy-based stability analysis. Different textbooks may define C differently while producing the same product Cq̇; individual entries of C are therefore not unique, but the physical dynamic model is equivalent when the complete equation is preserved.

6. Newton–Euler Formulation

The Newton–Euler method applies force and moment equations recursively to each link. It has two passes.

Outward Pass

Starting at the base, angular velocity, angular acceleration, and linear acceleration are propagated from link i−1 to link i. For a revolute joint, the joint velocity contributes to angular velocity; for a prismatic joint, it contributes to linear velocity.

Inward Pass

Starting at the end effector, forces and moments are propagated toward the base. The actuator effort is extracted by projecting the resulting wrench onto the joint axis. In symbolic form, each link obeys

where F_i and N_i are the resultant force and moment about the link center of mass. Newton–Euler recursion is computationally efficient and is widely used for real-time inverse dynamics. Euler–Lagrange is often more convenient for analytical derivation and energy reasoning.

Both methods should produce equivalent torques when the same frames, parameters, external loads, and sign conventions are used.

7. Forward and Inverse Dynamics

Inverse dynamics calculates torque from a prescribed motion:

It is used for feedforward control, actuator sizing, and trajectory evaluation.

Forward dynamics calculates acceleration from applied torque:

By rearranging the model,

The inverse M(q) exists for a physically valid configuration away from mechanical failure. Forward dynamics is used in simulation, model-based control, and digital commissioning.

8. Worked Dynamic Example: Single-Link Pendulum

Consider a uniform rigid link of mass m = 4 kg and length L = 0.60 m rotating in a vertical plane about one end. Let θ be measured from the upward vertical, and let positive torque increase θ. The center of mass is at L/2. The moment of inertia about the pivot is I_O = mL²/3.

The kinetic energy is

Taking gravitational potential energy as V = mg(L/2)cos θ, the Euler–Lagrange equation gives

Thus,

For θ = 60°, θ̈ = 2 rad/s², and g = 9.81 m/s²:

I_O = 4(0.60)²/3 = 0.48 kg·m².

τ_inertia = 0.48 × 2 = 0.96 N·m.

τ_gravity = 4 × 9.81 × 0.30 × sin 60° ≈ 10.20 N·m.

With the stated sign convention, τ ≈ 0.96 − 10.20 = −9.24 N·m. The negative sign means that the actuator must apply torque opposite the positive θ direction to maintain the specified upward-side acceleration under this coordinate convention. A different zero-angle definition changes the signs but not the physical requirement.

9. Actuator Sizing and Practical Design

A motor should not be selected using only static payload torque. The design torque may be estimated as

where S is a safety factor and τ_velocity represents Coriolis and centrifugal effects. This expression is a design decomposition rather than a universal independent-sum law; signs and phase relationships must be retained in a full dynamic calculation.

Important checks include:

• continuous RMS torque for thermal capacity;
• peak torque for acceleration and transient loads;
• motor speed and gearbox ratio;
• reflected inertia and gearbox efficiency;
• bearing and structural loads;
• brake torque for vertical axes;
• emergency-stop deceleration;
• singular configurations and workspace limits;
• encoder resolution and backlash.

For a gearbox with reduction ratio N and ideal efficiency, motor speed and load torque approximately satisfy ω_m = Nω_L and τ_m = τ_L/N. With efficiency η, the required motor torque is approximately τ_m = τ_L/(ηN) during motoring. Real gearboxes also introduce backlash, friction, compliance, and direction-dependent efficiency.

A common error is to compare a motor’s short-duration peak torque with a payload’s continuous gravitational requirement. Thermal limits, duty cycle, and repeated acceleration must be evaluated separately.

10. Applied Case: Payload Change in a Collaborative Robot

Background

Consider an illustrative six-axis collaborative robot used to place components into a fixture. Its controller was commissioned for a 2 kg tool and payload but is later asked to carry 4 kg.

Problem

The additional mass increases gravity torque at shoulder and elbow joints. During rapid placement, it also increases inertial torque. If the controller continues using the original payload model, position tracking may deteriorate and protective torque limits may be triggered.

Analysis

The payload wrench is mapped to joints through τ_payload = JᵀF. The gravity contribution changes with both payload mass and center-of-mass location. If the payload center is displaced from the flange, it creates a moment in addition to a force. Dynamic simulation should evaluate the complete trajectory, not merely one static pose, because J(q), M(q), and velocity-dependent terms vary throughout the motion.

Approach and Solution

The engineering team should:

1. Measure or obtain the tool and payload mass properties.
2. Update the robot’s payload mass and center-of-mass parameters.
3. Recalculate inverse dynamics along the planned trajectory.
4. Check peak and RMS joint torques, gearbox limits, and brake capacity.
5. Reduce acceleration or speed if required.
6. Validate the motion at low speed before production operation.
7. Confirm compliance with the robot manufacturer’s safety limits and applicable machinery and collaborative-robot standards.

Outcome

The updated model reveals that the shoulder joint exceeds its preferred continuous torque during a horizontal reach, while peak torque remains below the short-duration limit. The trajectory is modified to reduce acceleration and avoid the worst posture. The robot then operates within thermal and protective limits.

Learning Points

Payload rating is configuration-dependent. Mass alone is insufficient; center of mass, inertia tensor, speed, acceleration, duty cycle, and external contact all matter. Correct parameter identification is therefore a safety and productivity issue, not merely a simulation detail.

11. Limitations, Safety, and Contemporary Practice

Rigid-body models neglect link flexibility, joint compliance, cable forces, structural vibration, bearing deformation, gearbox hysteresis, and temperature-dependent parameters. At high speed or with lightweight arms, flexible-body and motor-electrical dynamics may be required. Friction models are especially difficult near zero velocity.

Dynamic models should be validated experimentally using safe trajectories and measured current, position, and acceleration. Identification methods can estimate masses, centers of mass, inertias, friction coefficients, and motor constants. Modern robot controllers increasingly combine analytical dynamics with data-driven residual models. Machine learning can compensate for unmodeled effects, but it should not replace physical safety limits, verified braking behavior, or deterministic protective functions.

Ethically and professionally, engineers must not exceed manufacturer payload or speed specifications merely because a simulation predicts acceptable torque. Safety-rated monitoring, risk assessment, guarding, emergency stops, and documented validation are essential. Any modification affecting payload, tooling, or software should be subject to change control and revalidation.

Best Practices

1. Define frames, positive directions, and units before deriving equations.
2. Use measured mass properties whenever possible.
3. Separate static, inertial, velocity-dependent, friction, and external-load contributions.
4. Check dimensions and signs at every stage.
5. Validate both Newton–Euler and Euler–Lagrange results on simple test cases.
6. Evaluate complete trajectories rather than isolated poses.
7. Distinguish peak, continuous, and RMS actuator requirements.
8. Include safety factors and account for uncertainty.
9. Validate models experimentally under controlled conditions.
10. Document payload assumptions and software parameter changes.

Chapter Summary

Robot statics determines equilibrium forces and torques using force and moment balance. The moment of a force is r × F, and ideal end-effector loads map to joint torques through τ = JᵀF. Payload mass and center of mass strongly affect gravity loading.

Robot dynamics includes translational and rotational inertia, gravity, Coriolis and centrifugal effects, friction, and external loads. The standard manipulator equation is M(q)q̈ + C(q,q̇)q̇ + g(q) + τ_f = τ + JᵀF_ext. Euler–Lagrange formulation derives this model from kinetic and potential energy, whereas Newton–Euler formulation uses recursive force and moment balances. Both are theoretically equivalent when assumptions and conventions match.

Inverse dynamics computes torque from motion and supports feedforward control and actuator sizing. Forward dynamics computes acceleration from applied torque and supports simulation. Practical design must consider peak and continuous torque, gearbox efficiency, inertia, thermal duty cycle, braking, structural limits, singularities, uncertainty, and safety requirements.

Key Terms

Statics: Analysis of bodies in equilibrium with zero acceleration.

Dynamics: Analysis relating motion, force, and torque.

Moment: The rotational effect of a force about a point or axis.

Center of mass: The mass-weighted average position of a body.

Payload: Tool, object, or material carried by a robot.

Jacobian: Matrix relating joint rates or displacements to Cartesian rates or displacements.

Virtual work: Work-based method for relating Cartesian forces to generalized joint forces.

Inertia: Resistance to angular acceleration.

Inertia tensor: Matrix describing rotational mass distribution in three dimensions.

Coriolis effect: Velocity-coupling torque involving products of different generalized velocities.

Centrifugal effect: Velocity-dependent torque commonly involving squared joint velocity.

Gravity vector: Generalized torque or force caused by gravitational potential.

Friction: Resistive force or torque associated with relative motion.

Euler–Lagrange method: Energy-based formulation of equations of motion.

Newton–Euler method: Recursive formulation based on force and moment balances.

Inverse dynamics: Calculation of actuator effort from prescribed motion.

Forward dynamics: Calculation of motion from applied actuator effort.

Singularity: Configuration in which the Jacobian loses rank and force or velocity transmission becomes problematic.

RMS torque: Root-mean-square torque used commonly for thermal-duty evaluation.

Payload identification: Estimation of mass properties needed for an accurate robot model.

Review Questions

Short Answer Questions

1. What two equilibrium equations describe a rigid body in static equilibrium?
2. Why does a payload center of mass affect joint torque?
3. What physical quantity does the inertia matrix represent?
4. State one principal difference between Newton–Euler and Euler–Lagrange formulations.
5. Why must peak and continuous motor torque be checked separately?

Long Answer Questions

1. Derive the relationship τ = JᵀF using virtual work and explain its assumptions.
2. Explain each term in the standard rigid-manipulator dynamic equation.
3. Compare Newton–Euler and Euler–Lagrange methods in terms of derivation, computation, and application.
4. Explain how friction, gearbox efficiency, and payload inertia influence actuator selection.
5. Discuss the limitations of rigid-body dynamics models in high-speed or lightweight robots.

Analytical / Application Questions

1. A planar arm carries a payload farther from its base without changing payload mass. Predict how the static base-joint torque changes and justify your answer using a moment equation.
2. A robot tracks a trajectory accurately at low speed but shows large errors at high speed. Identify the dynamic terms that may be responsible and propose a validation procedure.
3. Design a calculation and safety-validation workflow for increasing a collaborative robot payload from 2 kg to 4 kg, including mass-property identification, torque checks, thermal considerations, and operational validation.

Multiple Choice Questions with Answers

Question 1

Which condition defines static equilibrium of a rigid body?

A. The net force is zero only

B. The net moment is zero only

C. Both net force and net moment are zero

D. The velocity is constant but acceleration is nonzero

Correct Answer: C. Both net force and net moment are zero

Question 2

For an end-effector force F and Jacobian J, the ideal joint torque is:

A. τ = JF

B. τ = JᵀF

C. τ = J⁻¹F

D. τ = FᵀJᵀF

Correct Answer: B. τ = JᵀF

Question 3

What is the SI unit of a mass moment of inertia?

A. N

B. N·m

C. kg·m²

D. kg/s²

Correct Answer: C. kg·m²

Question 4

Which term represents gravity in the standard manipulator equation?

A. M(q)q̈

B. C(q,q̇)q̇

C. g(q)

D. JᵀF_ext

Correct Answer: C. g(q)

Question 5

The Euler–Lagrange method is primarily based on:

A. Kinetic and potential energy

B. Electrical resistance and voltage

C. Only static force balance

D. Gearbox efficiency measurements

Correct Answer: A. Kinetic and potential energy

Question 6

Inverse dynamics calculates:

A. Joint acceleration from torque

B. Torque from prescribed motion

C. Payload mass from voltage only

D. Cartesian position from gravity alone

Correct Answer: B. Torque from prescribed motion

Question 7

For a 5 kg payload producing a force at a perpendicular distance of 0.40 m, with g = 9.81 m/s², the static moment magnitude is approximately:

A. 4.91 N·m

B. 19.62 N·m

C. 49.05 N·m

D. 196.2 N·m

Correct Answer: B. 19.62 N·m

Question 8

Which matrix is symmetric and positive definite for a physically valid robot configuration?

A. The gravity vector

B. The Jacobian

C. The inertia matrix

D. The friction sign function

Correct Answer: C. The inertia matrix

Question 9

Why can a robot require very large joint torques near a singularity?

A. Gravity becomes zero

B. The Jacobian loses rank and force transmission becomes poorly conditioned

C. The links lose all mass

D. Friction always becomes infinite

Correct Answer: B. The Jacobian loses rank and force transmission becomes poorly conditioned

Question 10

Which actuator specification is most directly associated with long-term heating during repeated motion?

A. Continuous or RMS torque

B. Peak position error only

C. Payload color

D. Single instantaneous torque only

Correct Answer: A. Continuous or RMS torque