3. Robot Kinematics and Workspace Analysis

Introduction and Context

Robot kinematics is the study of motion without considering the forces and torques that cause that motion. In robotics, it provides the mathematical connection between joint variables and the position and orientation of a robot’s end-effector. This connection is central to simulation, trajectory planning, control, calibration, collision avoidance, and task programming.

A manipulator is commonly described as a serial or parallel arrangement of rigid links connected by joints. The joint variables—angles for revolute joints and displacements for prismatic joints—define the robot configuration. Kinematics answers two complementary questions. Forward kinematics determines the end-effector pose from known joint variables. Inverse kinematics determines the joint variables required to achieve a specified pose.

The problem is more than a coordinate transformation. A robot may have multiple inverse solutions, no solution, or infinitely many solutions. Its ability to generate motion also changes throughout the workspace. Near a singular configuration, certain Cartesian motions become difficult or impossible, while small Cartesian commands may demand very large joint velocities. Workspace analysis and Jacobian-based measures therefore provide essential tools for evaluating robot capability.

This chapter develops these ideas from homogeneous transformations and Denavit–Hartenberg modelling through velocity kinematics, singularities, manipulability, redundancy, workspace computation, and calibration errors. The discussion assumes familiarity with vectors, matrices, differential calculus, and basic rigid-body transformations.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Define robot configuration, pose, workspace, reachability, dexterity, and singularity.

2. Construct forward kinematic models using homogeneous transformations and Denavit–Hartenberg parameters.

3. Formulate and solve inverse kinematics using geometric, algebraic, numerical, and optimization-based methods.

4. Derive the Jacobian and use it to relate joint velocities to end-effector twists.

5. Identify and interpret singular configurations and their consequences for control and planning.

6. Evaluate manipulability, isotropy, redundancy, and feasible workspace.

7. Explain the effects of calibration errors and select appropriate validation and compensation methods.

8. Apply kinematic analysis to industrial robot selection, programming, maintenance, and safety-critical operation.

Configuration, Pose, and the Geometry of Motion

The configuration of an n-joint robot is represented by the vector

q = [q₁ q₂ … qₙ]ᵀ.

For a revolute joint, qᵢ is an angle; for a prismatic joint, it is a linear displacement. The configuration space, often denoted C-space, is the set of all physically permitted values of q. Joint limits, mechanical stops, cable routing, collision constraints, and actuator limits restrict the usable portion of this space.

The pose of a rigid body consists of position and orientation. Position is represented by a vector p = [x y z]ᵀ, while orientation is represented by a rotation matrix R ∈ SO(3), satisfying RᵀR = I and det(R) = 1. A homogeneous transformation combines both:

T = [ R p ; 0 1 ].

The transformation from frame B to frame A is written as ᴬTᴮ. Its upper-left 3 × 3 block gives the orientation of frame B relative to frame A, and its last column gives the origin of frame B expressed in frame A. Transformations compose through matrix multiplication:

ᴬTᶜ = ᴬTᴮ ᴮTᶜ.

The inverse transformation is

T⁻¹ = [ Rᵀ −Rᵀp ; 0 1 ].

This notation is useful because a serial robot’s pose is obtained by multiplying the transformations associated with its links and joints. Careful frame assignment is essential: many apparent kinematic errors are actually frame-convention errors.

Important Terminology

The base frame is fixed to the robot support, and the tool frame is attached to the end-effector. A link is a rigid structural member, whereas a joint permits relative motion between adjacent links. The task space is the space of end-effector positions and orientations. The workspace is the subset of task space that the robot can reach under specified constraints. Reachability refers to the existence of at least one configuration for a desired pose; dexterity additionally concerns the quality of motion and orientation control at that pose.

A configuration is collision-free if no robot link, tool, or payload intersects an obstacle or forbidden region. A kinematic solution is feasible only if it satisfies joint limits, mechanical constraints, and—when used in practice—velocity, acceleration, torque, and environmental requirements.

Forward Kinematics: From Joints to End-Effector Pose

Forward kinematics is deterministic once the robot geometry and joint configuration are specified. For a serial manipulator,

⁰Tₙ(q) = ⁰T¹(q₁) ¹T²(q₂) … ⁿ⁻¹Tⁿ(qₙ).

The result is a 4 × 4 homogeneous transformation containing the end-effector position and orientation. Forward kinematics is usually straightforward computationally, but its accuracy depends on the correctness of link dimensions, joint zero offsets, axis directions, and frame conventions.

Denavit–Hartenberg Representation

The classical Denavit–Hartenberg (DH) convention describes each link transformation using four parameters: link length aᵢ, link twist αᵢ, link offset dᵢ, and joint angle θᵢ. A commonly used standard DH transformation is

Aᵢ = Rot(z, θᵢ) Trans(z, dᵢ) Trans(x, aᵢ) Rot(x, αᵢ).

Its matrix form is

Aᵢ =

[ cθ −sθcα sθsα a cθ ;

sθ cθcα −cθsα a sθ ;

0 sα cα d ;

0 0 0 1 ],

where cθ = cos θᵢ, sθ = sin θᵢ, cα = cos αᵢ, and sα = sin αᵢ. For a revolute joint, θᵢ generally varies and dᵢ is constant; for a prismatic joint, dᵢ varies and θᵢ is constant.

The modified DH convention changes the order and location of elementary transformations. Both conventions are valid, but a model must not mix their parameter definitions. In software, the selected convention should be documented and tested using simple configurations.

Illustrative Two-Link Planar Manipulator

For a two-revolute planar arm with link lengths l₁ and l₂, joint angles q₁ and q₂, the end-effector position is

x = l₁ cos q₁ + l₂ cos(q₁ + q₂),

y = l₁ sin q₁ + l₂ sin(q₁ + q₂).

If the tool orientation is measured relative to the base x-axis, then

φ = q₁ + q₂.

This compact example reveals several general principles. Joint q₁ influences both links, while q₂ influences only the second link. The reachable position boundary has an outer radius l₁ + l₂ and, when l₁ > l₂, an inner radius |l₁ − l₂|. However, not every point within this annular region necessarily permits every orientation.

Inverse Kinematics: From Desired Pose to Joint Configuration

Inverse kinematics (IK) solves

f(q) = x_d,

where f is the forward kinematic function and x_d is the desired task-space pose. Unlike forward kinematics, IK may have multiple solutions or may be ill-conditioned. A solution must be checked against joint limits, collisions, self-intersections, tool constraints, and continuity with the preceding motion.

Geometric and Algebraic Solutions

Geometric methods exploit the structure of a manipulator. For the two-link planar arm, define r² = x² + y². Applying the cosine rule gives

cos q₂ = (r² − l₁² − l₂²)/(2l₁l₂).

Thus,

q₂ = ± cos⁻¹[(r² − l₁² − l₂²)/(2l₁l₂)],

representing the elbow-up and elbow-down configurations. The first joint can then be obtained from

q₁ = atan2(y,x) − atan2(l₂ sin q₂, l₁ + l₂ cos q₂).

The inverse cosine argument must lie between −1 and 1. If it does not, the target position is outside the reachable region. The two signs produce distinct branches, which may merge at a stretched or folded boundary configuration.

Algebraic methods extend this approach by eliminating variables from the transformation equations. They can provide all solutions for certain six-degree-of-freedom industrial arms, especially those with a spherical wrist. Their disadvantages include complex symbolic expressions, sensitivity to numerical round-off, and difficulty handling arbitrary mechanisms.

Numerical Inverse Kinematics

Numerical methods are more general. Given a current estimate qₖ, the task error eₖ is approximated locally by

eₖ ≈ J(qₖ) Δqₖ.

A basic update is

qₖ₊₁ = qₖ + J⁺eₖ,

where J⁺ is the Moore–Penrose pseudoinverse. Iteration continues until the position and orientation errors are sufficiently small. Numerical IK is suitable for robots with nonstandard geometry, closed constraints, or additional optimization objectives.

The choice of orientation error is important. Subtracting Euler angles can create discontinuities and depends on the selected sequence. Rotation-vector or quaternion-based errors are generally more robust. A practical error vector may be written as e = [eₚᵀ eᵣᵀ]ᵀ, where eₚ is position error and eᵣ is a three-dimensional rotational error.

Near singularities, the pseudoinverse may generate unacceptably large joint changes. Damped least squares replaces the pseudoinverse operation with

Δq = Jᵀ(JJᵀ + λ²I)⁻¹e,

where λ is a damping factor. Damping improves numerical stability, although it introduces a small residual task error. Adaptive damping can increase λ as the smallest singular value approaches zero.

Optimization-Based IK

In industrial applications, IK is often posed as a constrained optimization problem. A typical objective is

minimize ||e(q)||² + w₁||q − q₀||² + w₂C(q),

subject to joint limits, collision constraints, and other inequalities. The second term encourages continuity or proximity to a preferred configuration; C(q) may penalize low manipulability, proximity to joint limits, or energy consumption. This formulation is particularly valuable for redundant robots, where many configurations can accomplish the same task.

Jacobian and Velocity Kinematics

The Jacobian is the differential map between joint velocity and end-effector velocity. For an m-dimensional task,

ẋ = J(q) q̇.

For a spatial robot, the end-effector velocity is represented by a twist:

V = [v; ω],

where v is linear velocity and ω is angular velocity. Thus, for a six-dimensional task,

V = J(q)q̇,

with J typically a 6 × n matrix.

For a revolute joint whose axis zᵢ and origin oᵢ are expressed in the base frame, the Jacobian column is

Jᵢ = [ zᵢ × (oₙ − oᵢ) ; zᵢ ].

For a prismatic joint,

Jᵢ = [ zᵢ ; 0 ].

The linear part is called Jᵥ and the angular part Jω. This geometric Jacobian is often constructed directly from the forward-kinematic frames. An analytical Jacobian may instead map joint rates to derivatives of a particular orientation parameterization, such as Euler angles. The two are not identical because orientation coordinates have their own singularities.

Static Force Relationship

Under ideal virtual-work assumptions, joint torques and end-effector wrench are related by

τ = JᵀF,

where F contains forces and moments. This relation links velocity transmission to force transmission. A direction in which the robot has poor velocity capability may correspond to high force transmission, and vice versa. Practical actuator limits, friction, structural compliance, and controller bandwidth modify this ideal relationship.

Acceleration and Higher-Order Effects

Differentiating the velocity equation gives

ẍ = J(q)q̈ + J̇(q,q̇)q̇.

The second term represents velocity-dependent effects caused by changing joint axes and lever arms. Omitting J̇q̇ can produce substantial trajectory errors during fast motion. Kinematic acceleration relationships are used in model-based control, trajectory generation, and dynamic analysis, although forces and inertial effects belong formally to robot dynamics.

Singularities and Manipulability

A singularity occurs when the Jacobian loses rank. At such a configuration, one or more independent Cartesian velocity directions cannot be produced by finite joint velocities. For a square Jacobian, singularity is often detected by det(J) = 0. For a rectangular Jacobian, rank deficiency or a zero singular value is the appropriate criterion.

The singular value decomposition,

J = UΣVᵀ,

provides a particularly informative interpretation. The columns of V identify joint-space directions, while the columns of U identify Cartesian directions. The singular values in Σ describe velocity scaling in those directions. A very small singular value indicates a poorly controllable Cartesian direction and predicts large joint rates for a desired velocity in that direction.

Common singularity types include:

• Boundary or arm singularities, often occurring when links become collinear or fully stretched.
• Wrist singularities, in which two rotational axes align and one orientation degree of freedom is lost.
• Internal or alignment singularities caused by special geometric relationships among axes.

Singularities can cause joint-speed spikes, numerical instability, loss of directional control, abrupt configuration changes, and excessive actuator demand. They should not be treated merely as mathematical inconveniences. A trajectory that passes close to a singularity may be unacceptable even if it does not encounter an exact rank loss.

Manipulability Measures

Yoshikawa’s velocity manipulability measure is

w(q) = √det(JJᵀ),

for a full-row-rank Jacobian. It is proportional to the volume of the velocity ellipsoid generated by unit-norm joint velocities. A value approaching zero indicates a singular or nearly singular configuration.

The condition number, κ = σmax/σmin, measures isotropy. Low values indicate more uniform transmission of motion, while a large value indicates directional weakness. Because translational and rotational quantities have different units, a combined six-dimensional measure requires suitable scaling. In practice, task-specific weighting is preferable to blindly interpreting an unscaled determinant.

The manipulability ellipsoid is defined by

vᵀ(JJᵀ)⁻¹v = 1

for a velocity-only Jacobian of full row rank. Its principal axes show directions of strong and weak motion transmission. The corresponding force ellipsoid is inversely related under ideal assumptions. These visual and numerical tools help planners select configurations that preserve dexterity during assembly, welding, insertion, and inspection.

Redundancy and Null-Space Motion

A robot is kinematically redundant for a task when it has more joint degrees of freedom than the task requires. For example, a seven-joint manipulator performing a six-dimensional pose task has one degree of redundancy. Redundancy can improve obstacle avoidance, joint-limit avoidance, posture selection, and manipulability, but it makes IK nonunique.

The general velocity solution is

q̇ = J⁺ẋ + (I − J⁺J)z,

where z is an arbitrary vector. The first term achieves the desired task velocity; the second lies in the Jacobian null space and produces no instantaneous task motion. A secondary objective can be introduced through a scalar function h(q), such as distance from joint limits. A common choice is

z = k∇h(q),

with sign selected so that h improves. Null-space motion must be implemented carefully because finite motions, obstacles, and model errors can cause task deviations even when instantaneous velocity is zero.

Redundancy resolution is therefore a decision problem. A posture that maximizes manipulability may not be safest, most energy-efficient, or best for cable routing. Industrial controllers commonly prioritize hard constraints first and optimize softer objectives second.

Workspace Analysis

The workspace is not a single universal region; it depends on what is held fixed and which constraints are included. Position workspace describes reachable origins of the tool frame, while orientation workspace describes attainable orientations at a position. The dexterous workspace is the set of positions at which all required orientations can be achieved. The reachable workspace may include positions attainable in only one orientation or one configuration branch.

Joint limits, tool geometry, payload, collision restrictions, singularity avoidance, and environmental access all reduce the theoretical workspace. For a planar two-link arm, the annular region gives a simple analytical position workspace. For spatial six-axis robots, analytical descriptions may be difficult, and computational methods are more practical.

Workspace Computation Methods

Analytical methods derive boundary equations from geometry. They are valuable for simple mechanisms and for understanding how link lengths affect reach, but they become cumbersome for general robots.

Sampling methods select joint configurations within permitted limits, compute forward kinematics, and accumulate the resulting points or poses. Random Monte Carlo sampling is easy to implement and handles irregular constraints, although it can miss thin or disconnected regions. Grid-based sampling gives more uniform coverage but may be computationally expensive.

A voxel or occupancy representation divides position space into small cells and marks cells reached by valid configurations. Orientation can be represented by bins, quaternions, or task-specific orientation tests. Boundary-based methods estimate the outer envelope, while implicit methods use optimization to test whether a target is reachable.

A robust workspace study should distinguish theoretical reachability from operational usability. A point close to a reach boundary may be reachable only at a singularity, with poor load capacity or very low tolerance to calibration error. Consequently, production planning should use a safety margin around workspace boundaries and singularity regions.

Calibration and Kinematic Accuracy

A robot’s nominal kinematic model rarely matches its physical geometry exactly. Sources of error include link-length deviations, joint-axis misalignment, joint zero offsets, encoder quantization, gearbox backlash, structural deformation, thermal expansion, tool-center-point errors, and inaccurate base-frame placement. These errors may be small in individual parameters but significant at the tool tip, particularly when long lever arms amplify angular errors.

A calibrated model can be written as

x = f(q, p),

where p contains geometric and nongeometric parameters. Calibration estimates p from measured poses. The robot is commanded to a set of configurations, and the predicted tool pose is compared with measurements obtained from a laser tracker, photogrammetry system, coordinate-measuring machine, or calibrated vision system. Parameter estimation may use least squares:

p̂ = arg min Σᵢ ||xᵢ,measured − f(qᵢ,p)||².

The experiment must be designed so that parameters are identifiable. If all measurements are taken in similar postures, different parameter errors may produce nearly identical effects and cannot be separated reliably. Configurations should excite different joint axes and workspace regions. Validation must use independent poses rather than only the data used for fitting.

Calibration should also clarify the error metric. Position error may be measured in millimetres, while orientation error may be expressed as an angle or rotation-vector norm. Combining them requires weighting consistent with the application. Repeatability and absolute accuracy are different: a robot may return repeatedly to the same incorrect location, showing good repeatability but poor accuracy.

Applied Case: Calibration and Singularity-Aware Planning for a Robotic Assembly Cell

Background

Consider an illustrative automotive assembly cell using a six-axis robot to insert a precision component into a housing. The task requires a tool-center-point accuracy of approximately a fraction of a millimetre, smooth approach along the insertion axis, and avoidance of contact with the housing walls.

Situation and Problem

The robot reaches the nominal insertion pose according to its controller, but the component intermittently contacts the housing. Operators also report sudden increases in wrist-joint speed during the final approach. Initial inspection shows that the target lies close to a wrist singularity, while the robot’s tool transformation and joint zero offsets have not been verified since the end-effector was replaced.

Analysis

The forward model predicts a correct nominal pose, but measurement of several independent configurations reveals a systematic position error that changes with posture. This pattern indicates geometric or frame-calibration errors rather than random repeatability error. The Jacobian at the insertion posture has a very small minimum singular value. Consequently, a small desired Cartesian rotation about one direction requires a disproportionately large combination of wrist joint velocities.

The direct use of a pseudoinverse magnifies measurement noise and command discretization. In addition, the insertion path was planned as a straight Cartesian line without checking the singularity margin or joint-speed limits.

Approach and Solution

First, the tool-center-point transformation is measured and entered into the robot model. Joint zero offsets and selected link parameters are estimated from a set of well-distributed calibration poses. The fitted model is validated on independent poses.

Second, the insertion trajectory is re-planned using damped least-squares inverse kinematics. The planner adds a penalty for low minimum singular value and for proximity to joint limits. A nearby redundant posture is not available because the robot has exactly six joints, so the approach orientation is modified within the allowable assembly tolerance to move the wrist away from alignment.

Third, a low-speed guarded approach is introduced. The controller monitors joint velocity, Cartesian error, and motor current. The process stops if contact is detected outside the permitted insertion stage.

Outcome

The calibrated model reduces systematic tool error, while the new path avoids the near-singular posture and limits wrist speed. The guarded approach adds a small cycle-time cost but reduces the risk of component damage and protects the tooling.

Learning Points

The case illustrates that kinematic accuracy, singularity avoidance, and process safety are interconnected. A mathematically reachable pose is not necessarily an operationally suitable pose. Calibration must be validated independently, and trajectory generation should consider Jacobian conditioning rather than position error alone.

Practical and Professional Applications

Kinematic analysis supports robot selection by indicating whether a candidate robot can reach required points with adequate orientation freedom and dexterity. In layout design, workspace models help determine mounting height, allowable part presentation, and accessibility to fixtures. In programming, multiple IK branches must be managed to prevent unexpected elbow or wrist flips. In maintenance, deviations in repeatability or pose accuracy can indicate loose tooling, encoder problems, or mechanical wear.

Digital twins and offline programming systems use the kinematic model to simulate reachability, cycle times, collision risk, and coordinated motion. Vision-guided robots combine camera-based object poses with inverse kinematics, making frame registration and calibration especially important. Optimization and machine-learning methods may assist in selecting configurations or estimating residual errors, but learned models should be checked against physical constraints and verified over the intended operating envelope.

Professional practice requires traceability of coordinate frames, parameter versions, calibration dates, and software changes. A kinematic model used for safety-related motion should not be altered without controlled validation. Human–robot collaboration further requires conservative treatment of model uncertainty, unexpected contact, and degraded sensor conditions.

Challenges, Limitations, and Common Errors

Several recurring mistakes compromise kinematic analysis:

• Mixing standard and modified DH conventions.
• Assigning joint axes inconsistently or reversing a positive rotation direction.
• Confusing a tool pose with the flange pose.
• Treating Euler-angle differences as a globally valid orientation error.
• Selecting an IK solution without checking joint limits or collision constraints.
• Using a pseudoinverse near a singularity without damping or speed supervision.
• Comparing manipulability values from differently scaled Jacobians.
• Assuming a reachable workspace is equally usable everywhere.
• Calibrating with poorly distributed poses and reporting only fitting error.
• Treating nominal geometry as exact after tooling, temperature, or payload changes.

Numerical methods also have limitations. They depend on initial guesses, may converge to an undesirable branch, and can fail near discontinuities or constraints. Sampling-based workspace estimates depend on resolution and may provide misleading confidence in sparsely sampled regions. Jacobian-based analysis is local; it describes instantaneous motion and does not by itself guarantee a collision-free finite trajectory.

Best Practices for Kinematic Modelling and Analysis

A reliable workflow begins with an explicit frame diagram and a documented joint convention. Forward kinematics should be verified at zero, limiting, and easily interpretable configurations. Independent implementations or simulation tools can be used for cross-checking, but the underlying frame definitions must agree.

For inverse kinematics, use analytical solutions when they are reliable and computationally justified; otherwise use numerical or optimization-based methods with meaningful initial guesses. Rank, singular values, joint limits, and collision constraints should be monitored continuously during trajectory execution.

For workspace studies, report whether the result concerns position, full pose, dexterous reachability, or constrained operational reachability. Include tool geometry and relevant environmental restrictions. For calibration, use identifiable parameter sets, independent validation poses, and application-specific accuracy metrics. Finally, preserve a safety margin: boundary configurations, poorly conditioned postures, and uncertain calibration should not be used as ordinary production configurations.

Current Developments and Emerging Directions

Modern robot systems increasingly combine classical kinematics with real-time optimization, perception, and adaptive calibration. Redundant collaborative robots use null-space control to maintain comfortable or safe postures while following task trajectories. Model-predictive methods can incorporate future singularity risk, obstacle motion, and actuator limits rather than reacting only to instantaneous errors.

Online calibration techniques estimate changes caused by tool replacement, thermal growth, or structural deflection. Digital twins can compare commanded and measured motion to detect model drift. Data-driven residual models may improve accuracy, but they should supplement rather than replace physically interpretable kinematic models. Emerging approaches also integrate uncertainty propagation, allowing planners to select configurations that are not only reachable but robust to parameter and sensing errors.

Chapter Summary

Robot kinematics establishes the relationship between joint configuration and end-effector pose. Homogeneous transformations and DH parameters provide systematic forward models, while inverse kinematics may be solved geometrically, algebraically, numerically, or through constrained optimization. The Jacobian maps joint velocities to Cartesian twists and also relates end-effector wrenches to joint torques through virtual work.

Singularities are rank-deficient configurations in which motion capability is lost or poorly conditioned. Singular values, condition numbers, manipulability ellipsoids, and Yoshikawa’s measure provide tools for assessing dexterity. Redundant robots exploit null-space motion to satisfy secondary objectives such as obstacle and joint-limit avoidance. Workspace analysis must distinguish theoretical reachability from practical, orientation-constrained, collision-free operation. Finally, calibration identifies discrepancies between nominal and physical geometry; accurate modelling, independent validation, and controlled implementation are essential for dependable industrial performance.

Key Terms

Configuration: A complete set of joint variables describing the robot posture.

Configuration space: The set of all permitted robot joint configurations.

Forward kinematics: Determination of end-effector pose from joint variables.

Inverse kinematics: Determination of joint variables for a desired end-effector pose.

Homogeneous transformation: A 4 × 4 matrix representing rigid-body rotation and translation.

Denavit–Hartenberg parameters: Four parameters used to describe successive robot-link transformations.

Pose: The combined position and orientation of a rigid body.

Task space: The space of variables describing the desired end-effector task.

Workspace: The set of task-space locations or poses attainable by a robot.

Reachability: The existence of at least one feasible configuration for a target pose.

Dexterity: The quality and directional capability of motion at a configuration.

Jacobian: The differential matrix mapping joint rates to end-effector velocity.

Twist: A vector combining translational and angular velocity.

Singularity: A configuration at which the Jacobian loses rank.

Manipulability: A measure of how effectively joint motion produces Cartesian motion.

Redundancy: Having more joint degrees of freedom than required by a task.

Null space: Joint-velocity directions that produce no instantaneous task velocity.

Calibration: Estimation and compensation of differences between nominal and physical robot parameters.

Repeatability: The ability to return consistently to the same location.

Absolute accuracy: The closeness of a robot’s achieved pose to the commanded pose.

Review Questions

Short Answer Questions

1. What information is contained in a homogeneous transformation matrix?

2. Distinguish between reachable workspace and dexterous workspace.

3. Why can inverse kinematics have more than one solution?

4. What physical meaning is associated with a small singular value of the Jacobian?

5. State the difference between repeatability and absolute accuracy.

Long Answer Questions

1. Explain the forward kinematic modelling procedure using the Denavit–Hartenberg convention.

2. Compare geometric, analytical, numerical, and optimization-based inverse kinematics methods.

3. Derive the geometric Jacobian columns for revolute and prismatic joints and explain their meaning.

4. Discuss singularities, their classification, their consequences, and methods for avoiding them.

5. Explain how robot calibration is performed and why experimental pose selection affects parameter identifiability.

Analytical / Application Questions

1. For a two-link planar robot, derive the inverse kinematic equations and identify the condition under which no real solution exists.

2. A six-axis robot exhibits very large wrist velocities during a straight Cartesian approach. Explain how singular-value analysis could diagnose the problem and propose a revised control method.

3. Design a workspace-analysis procedure for a robot operating around fixtures, including joint limits, collision constraints, orientation requirements, and validation of the computed workspace.

22. Multiple Choice Questions with Answers

Question 1

For a serial robot, which expression represents the forward kinematic transformation from the base to the end-effector?

A. The sum of all joint variables

B. The product of successive link transformations

C. The inverse of the Jacobian multiplied by the joint vector

D. The determinant of the transformation matrices

Correct Answer: B. The product of successive link transformations

Question 2

For a two-link planar revolute arm, the two signs in the inverse solution for q₂ generally represent which configurations?

A. Prismatic and revolute configurations

B. Base and tool frames

C. Elbow-up and elbow-down configurations

D. Singular and nonsingular configurations

Correct Answer: C. Elbow-up and elbow-down configurations

Question 3

What does the equation ẋ = J(q)q̇ describe?

A. The static force relationship

B. The mapping from joint velocity to task-space velocity

C. The relationship between link mass and acceleration

D. The workspace boundary equation

Correct Answer: B. The mapping from joint velocity to task-space velocity

Question 4

Which condition most directly indicates a Jacobian singularity?

A. All joint angles are zero

B. The Jacobian has full rank

C. At least one singular value of the Jacobian is zero

D. The robot has more joints than task dimensions

Correct Answer: C. At least one singular value of the Jacobian is zero

Question 5

Why is damped least-squares inverse kinematics useful near a singularity?

A. It eliminates the need for a forward model

B. It limits the amplification of joint velocities caused by poor conditioning

C. It guarantees the exact target pose in one iteration

D. It converts all revolute joints into prismatic joints

Correct Answer: B. It limits the amplification of joint velocities caused by poor conditioning

Question 6

In the velocity solution q̇ = J⁺ẋ + (I − J⁺J)z, what is the role of the second term?

A. It produces only gravitational compensation

B. It generates null-space motion that does not instantaneously affect the task

C. It calculates the end-effector orientation directly

D. It removes all joint limits

Correct Answer: B. It generates null-space motion that does not instantaneously affect the task

Question 7

Which statement best distinguishes repeatability from absolute accuracy?

A. Repeatability concerns speed, whereas accuracy concerns torque

B. Repeatability is consistency of return, whereas accuracy is closeness to the commanded pose

C. Repeatability applies only to prismatic robots

D. Accuracy is independent of calibration

Correct Answer: B. Repeatability is consistency of return, whereas accuracy is closeness to the commanded pose

Question 8

A point lies within a robot’s position workspace but cannot be reached with the required tool orientation. What does this demonstrate?

A. Position reachability does not guarantee full-pose reachability

B. The Jacobian must have six rows for every robot

C. The robot has no inverse kinematic solution at any point

D. The point is necessarily a collision state

Correct Answer: A. Position reachability does not guarantee full-pose reachability

Question 9

Which calibration practice improves parameter identifiability?

A. Using measurements from only one repeated posture

B. Selecting configurations that excite different joint axes and workspace regions

C. Ignoring tool-center-point errors

D. Validating only on the poses used for fitting

Correct Answer: B. Selecting configurations that excite different joint axes and workspace regions

Question 10

For an ideal manipulator, the relation τ = JᵀF connects which quantities?

A. Joint torques and end-effector wrench

B. Joint positions and link lengths

C. Workspace volume and calibration error

D. Orientation coordinates and Euler-angle rates

Correct Answer: A. Joint torques and end-effector wrench