Practice exam

Question 1Not yet answeredMarked out of 3.00Differential kinematics

Which of the following does forward differential kinematics map?

Cartesian velocities of the end-effector into joint velocitiesEnd-effector forces into joint torquesJoint velocities into Cartesian velocities of the end-effectorJoint torques into end-effector forces

Question 2Not yet answeredMarked out of 3.00Product of exponentials

Which of the following does the product of exponentials depend on?

Material of the robotInitial temperature of the robotInitial configuration of the end-effectorWeight of the robot

Question 3Not yet answeredMarked out of 3.00Orthonormal basis

Together

e_x

e_y

, and

e_z

form an orthonormal basis. What is the vector

e_y \times e_z

-e_x

e_x

-e_y

e_z

Question 4Not yet answeredMarked out of 3.00Inverse kinematics

Consider a robotic arm with six joints. From which group is the result of the inverse kinematics an element?

\mathbb{R}^6

\mathbb{R}^2

\mathbb{R}^3

\mathbb{R}

Question 5Not yet answeredMarked out of 3.00Inverse kinematics

Which of the following is true for inverse kinematics?

It transforms joint positions into end-effector pose.It transforms end-effector pose into joint positions.It transforms end-effector velocities into joint velocities.It transforms joint velocities into end-effector velocities.

Question 6Not yet answeredMarked out of 3.00Forward kinematics

Consider a robot with forward kinematics

\begin{bmatrix} x_e \\ y_e \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} q_1 \\ q_2 \end{bmatrix}

. Which configuration reaches end-effector position

x_e = 6

y_e = 2

q_1 = 0,\ q_2 = 2

q_1 = 0,\ q_2 = 6

q_1 = 6,\ q_2 = 0

q_1 = 2,\ q_2 = 0

Question 7Not yet answeredMarked out of 3.00Singularities

The current configuration of a robot has led to the Jacobian matrix

J = \begin{bmatrix} 1 & 3 \\ 3 & 9 \end{bmatrix}

. This configuration is ...

a singular one.a synchronous one.a revolute one.a flexible one.

Question 8Not yet answeredMarked out of 3.00Statics

What would we use to map external forces to joint torques while the robot is at rest?

The inverse transposed JacobianThe transposed JacobianThe manipulability ellipsoidThe squared Jacobian

Question 9Not yet answeredMarked out of 3.00Workspace

What is the dimension of the workspace of a planar robot with seven joints (not considering the end-effector orientation)?

Exactly 6Exactly 2Exactly 7Exactly 3

Question 10Not yet answeredMarked out of 3.00Degrees of freedom

The four joints of a robot are denoted

q_1, \dots, q_4

. They are constrained by

q_1 + q_2 + q_3 + q_4 = 2\pi

. How many degrees of freedom does the robot have?

Exactly 2Exactly 4Exactly 1Exactly 3

Question 11Not yet answeredMarked out of 3.00DH convention

Which homogeneous transformation matrix corresponds to the DH parameters

d = 2\,\mathrm{m}

\theta = 90^\circ

T = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{bmatrix}

T = \begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}

T = \begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}

T = \begin{bmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{bmatrix}

Question 12Not yet answeredMarked out of 3.00Quaternions

Which quaternion corresponds to a rotation by angle

\theta = \pi

about a fixed axis with direction vector

w = [1\ 0\ 0]^T

Q = [0\ 1\ 0\ 0]^T

Q = [1\ 0\ 0\ 0]^T

Q = \left[\tfrac{1}{\sqrt{2}}\ \tfrac{1}{\sqrt{2}}\ 0\ 0\right]^T

Q = [1\ 1\ 0\ 0]^T

Question 13Not yet answeredMarked out of 3.00Singularities

Consider a robot with Jacobian

J = \begin{bmatrix} \cos(q_1+q_2) & \sin(q_1-q_2) \\ -\sin(q_1+q_2) & \cos(q_1-q_2) \end{bmatrix}

and

q_1 = 0

. Which value of

q_2

leads to a singular configuration?

q_2 = \tfrac{\pi}{4}

q_2 = \pi

q_2 = \tfrac{\pi}{2}

q_2 = 0

Question 14Not yet answeredMarked out of 3.00Trajectory planning

Which of the following statements concerning the jerk is not correct?

It impacts the energy consumption.It impacts the smoothness of joint positions.It impacts the joint actuators.It impacts the maximum velocity.

Question 15Not yet answeredMarked out of 6.00Rigid body pose

The point

d = [2\ 0\ 2]^T

of a rigid body, expressed in the body frame, has coordinates

d^* = [-2\ 0\ -2]^T

in the world frame. The orientation of the body with respect to the world frame is

R = \begin{bmatrix} 0 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}

. State the position of the body origin

r_C

. Provide the equation and the result.

Question 16Not yet answeredMarked out of 6.00Manipulator Jacobian

The relationship between joint configuration

q_1, q_2

and end-effector position is

x_e = 2 q_1 q_2 - q_1

and

y_e = q_1^2 - q_2

. Calculate the Manipulator Jacobian

J

Question 17Not yet answeredMarked out of 8.00Inverse differential kinematics

State the joint velocities

\dot{q}

needed to deliver the end-effector velocity

\dot{X} = [1\ 1]^T

with the Jacobian

J = \begin{bmatrix} 5 & 0 \\ 0 & 3 \end{bmatrix}

. Provide the equation and the result values.

Question 18Not yet answeredMarked out of 8.00Statics

Consider a robot with forward kinematics

\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} q_1 \\ q_2 \end{bmatrix}

. Compute the joint forces corresponding to the end-effector force

F_e = [0\ {-1}]^T

. Provide the equation and the result values.

Question 19Not yet answeredMarked out of 10.00Trajectory planning

Plan a point-to-point trajectory for a revolute joint. The start and end conditions are

q_0 = q(0) = -\pi\,\mathrm{rad}

and

q_f = q(T) = \pi\,\mathrm{rad}

, with duration

T = 4\,\mathrm{s}

. (a) State the general equation

q(t)

for a polynomial of the minimally required degree. (b) State the two equations from plugging in the conditions. (c) Calculate the coefficients.

Question 20Not yet answeredMarked out of 10.00Velocity from exponential coordinates

A rigid body rotates about the axis

w = [1\ 0\ 0]^T

at an offset

r_C = [0\ 0\ 1]^T

from the world frame (the body-frame origin sits at

r_C = [0\ 0\ 1]^T

). All vectors are in the world frame and the body is not translating. The velocity term in the exponential coordinate representation is

v = -\hat{w}\, r_C = -(w \times r_C)

. (a) Explain the symbol

v

by relating it to

\dot{r}_C

(which is zero, since the body does not translate). (b) Plug the vectors in and compute

v

You are allowed to use that.

Unit 1Configuration and DOF

Each revolute joint is a circle $S^1$ ; $n$ of them form the $n$ -torus.

Topology, not just dimension, is what distinguishes configuration spaces.

A free body in space has 6 dof: three to translate, three to rotate.

In the plane it drops to 3.

$m = 6$ for spatial mechanisms, $m = 3$ for planar ones.

$N$ counts links including ground, $J$ the joints, $f_i$ the dof of joint $i$ .

A single holonomic constraint between joints removes one dof.

Seen in Practice exam 3872.

A planar robot reaches a 2-D set of positions, no matter how many joints it has.

Extra joints add redundancy, not workspace dimension.

Seen in Practice exam 3872.

Unit 2Rigid Body Motions

The right-handed cyclic order $x \to y \to z \to x$ .

Reversing the order flips the sign.

Seen in Practice exam 3872.

Packs rotation and translation into one $4\times4$ matrix that composes by multiplication.

Not the plain transpose; the translation becomes $-R^{T} r$ .

Turns the exponential coordinates $\hat{w}\theta$ back into a rotation matrix.

World position = rotate the body-frame position, then shift by the origin $r_C$ .

Solve for the body origin with $r_C = p^{W} - R\, p^{B}$ .

Seen in Practice exam 3872.

The column pattern repeats for $R_x$ and $R_y$ on their own axes.

These two conditions define $SO(3)$ .

Orthonormal columns give $R^{-1} = R^{T}$ , so inverting a rotation is free.

The $3\times3$ matrix $[w]$ turns a cross product into a matrix product.

A rotation by angle $\theta$ about unit axis $w$ .

For $\theta = \pi$ the scalar part vanishes and $Q = [0, \ w^{T}]^{T}$ .

Seen in Practice exam 3872.

Unit 3Forward Kinematics

Four parameters per joint: twist $\alpha$ , length $a$ , offset $d$ , angle $\theta$ .

With $\alpha = 0$ , a pure $\theta$ rotation sits in the top-left $R_z$ block and $d$ shifts along $z$ .

Seen in Practice exam 3872.

Maps joint coordinates $q$ to the end-effector pose $x$ .

Always single-valued: one configuration gives one pose.

Seen in Practice exam 3872.

One screw axis $\mathcal{S}_i$ per joint, no chain of link frames needed.

$M$ is the home configuration of the end-effector, at $\theta = 0$ .

Seen in Practice exam 3872.

Angular part $\omega$ on top, linear part $v$ below.

For a pure rotation about an axis through point $q$ , $v = -\omega \times q$ .

Unit 4Inverse Kinematics

Turns a desired end-effector pose into joint coordinates.

For an $n$ -joint arm the solution lives in $\mathbb{R}^{n}$ ; it may have many or no solutions.

Seen in Practice exam 3872.

Iterate the Jacobian until the pose error shrinks below tolerance.

Uses the pseudo-inverse when $J$ is not square.

Unit 5Differential Kinematics and Statics

Joint velocities needed to deliver a wanted end-effector velocity.

Blows up near singularities, where $J^{-1}$ does not exist.

Seen in Practice exam 3872.

A square Jacobian loses rank; the arm cannot move in some direction.

Rows or columns become linearly dependent (e.g. one is a multiple of another).

Seen in Practice exam 3872.

How far the configuration is from a singularity; zero at one.

It is the volume of the manipulability ellipsoid.

Forward differential kinematics: joint velocities map to end-effector velocity.

Each column is the partial derivative of the pose with respect to one joint.

Seen in Practice exam 3872.

The transposed Jacobian maps an end-effector force to joint torques at rest.

It is the duality partner of $\dot{X} = J\dot{q}$ .

Seen in Practice exam 3872.

Unit 6Trajectory Planning

Four coefficients match position and velocity at both ends.

Smooth velocity, but the acceleration jumps at the endpoints.

The rate of change of acceleration; high jerk stresses actuators and wastes energy.

It affects smoothness, not the maximum velocity.

Seen in Practice exam 3872.

Two position boundary conditions need a degree-one polynomial, so two coefficients.

Plug in $q(0)$ and $q(T)$ to solve for $a_0$ and $a_1$ .

Seen in Practice exam 3872.

The linear part of the screw for a body rotating about an axis through $r_C$ .

It is the velocity of the body point momentarily at the world origin.

Seen in Practice exam 3872.

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Practice exam

Practice exam

Formulary

Unit 1Configuration and DOF

Unit 2Rigid Body Motions

Unit 3Forward Kinematics

Unit 4Inverse Kinematics

Unit 5Differential Kinematics and Statics

Unit 6Trajectory Planning

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