Mechanics: Kinematics formulary

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.

Unit 1Configuration and DOF

Configuration space of revolute jointsC=S1×⋯×S1⏟n=Tn\mathcal{C} = \underbrace{S^1 \times \cdots \times S^1}_{n} = T^nC=nS1×⋯×S1​​=Tn

Degrees of freedom of a rigid body3Ddof=6,2Ddof=33D dof = 6, \qquad 2D dof = 33Ddof=6,2Ddof=3

Grübler mobility formula(shows up on exams)dof=m (N−1−J)+∑i=1Jfi\mathrm{dof} = m\,(N - 1 - J) + \sum_{i=1}^{J} f_idof=m(N−1−J)+i=1∑J​fi​

Planar position workspace(shows up on exams)position workspace⊆R2\text{position workspace} \subseteq \mathbb{R}^2position workspace⊆R2

Unit 2Rigid Body Motions

Cross product of the basis(shows up on exams)ex×ey=ez,ey×ez=ex,ez×ex=eye_x \times e_y = e_z, \quad e_y \times e_z = e_x, \quad e_z \times e_x = e_yex​×ey​=ez​,ey​×ez​=ex​,ez​×ex​=ey​

Homogeneous transformT=[Rr03T1]T = \begin{bmatrix} R & r \\ 0_3^{T} & 1 \end{bmatrix}T=[R03T​​r1​]

Inverse homogeneous transformT−1=[RT−RTr03T1]T^{-1} = \begin{bmatrix} R^{T} & -R^{T} r \\ 0_3^{T} & 1 \end{bmatrix}T−1=[RT03T​​−RTr1​]

Matrix exponential of a rotation (Rodrigues)e[w^]θ=I+sin⁡θ [w^]+(1−cos⁡θ) [w^]2e^{[\hat{w}]\theta} = I + \sin\theta\,[\hat{w}] + (1 - \cos\theta)\,[\hat{w}]^{2}e[w^]θ=I+sinθ[w^]+(1−cosθ)[w^]2

Pose of a point(shows up on exams)pW=R pB+rCp^{W} = R\, p^{B} + r_CpW=RpB+rC​

Rotation about zRz(θ)=[cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001]R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}Rz​(θ)=​cosθsinθ0​−sinθcosθ0​001​​

Rotation matrix properties (SO(3))RTR=I,det⁡R=+1R^{T} R = I, \qquad \det R = +1RTR=I,detR=+1

Skew-symmetric matrix[w] x=w×x,[w]=−[w]T[w]\, x = w \times x, \qquad [w] = -[w]^{T}[w]x=w×x,[w]=−[w]T

Unit quaternion(shows up on exams)Q(θ,w)=[cos⁡θ2, sin⁡θ2 wT]TQ(\theta, w) = \left[\cos\tfrac{\theta}{2}, \ \sin\tfrac{\theta}{2}\, w^{T}\right]^{T}Q(θ,w)=[cos2θ​, sin2θ​wT]T

Unit 3Forward Kinematics

Forward kinematics(shows up on exams)x=f(q)x = f(q)x=f(q)

Product of exponentials(shows up on exams)T(θ)=e[S1]θ1 e[S2]θ2⋯e[Sn]θn MT(\theta) = e^{[\mathcal{S}_1]\theta_1}\, e^{[\mathcal{S}_2]\theta_2} \cdots e^{[\mathcal{S}_n]\theta_n}\, MT(θ)=e[S1​]θ1​e[S2​]θ2​⋯e[Sn​]θn​M

Screw axisS=[ωv]∈R6\mathcal{S} = \begin{bmatrix} \omega \\ v \end{bmatrix} \in \mathbb{R}^{6}S=[ωv​]∈R6

Unit 4Inverse Kinematics

Inverse kinematics(shows up on exams)q=f−1(x),q∈Rnq = f^{-1}(x), \qquad q \in \mathbb{R}^{n}q=f−1(x),q∈Rn

Numerical inverse kinematics (Newton step)qk+1=qk+J−1(qk) (xd−f(qk))q_{k+1} = q_k + J^{-1}(q_k)\,\bigl(x_d - f(q_k)\bigr)qk+1​=qk​+J−1(qk​)(xd​−f(qk​))

Unit 5Differential Kinematics and Statics

Inverse differential kinematics(shows up on exams)q˙=J−1 X˙\dot{q} = J^{-1}\, \dot{X}q˙​=J−1X˙

Kinematic singularity(shows up on exams)det⁡J=0\det J = 0detJ=0

Manipulability measurew=det⁡ ⁣(JJT)w = \sqrt{\det\!\left(J J^{T}\right)}w=det(JJT)​

Manipulator Jacobian(shows up on exams)X˙=J(q) q˙,J=∂x∂q\dot{X} = J(q)\,\dot{q}, \qquad J = \frac{\partial x}{\partial q}X˙=J(q)q˙​,J=∂q∂x​

Statics: force to torque(shows up on exams)τ=JTF\tau = J^{T} Fτ=JTF

Unit 6Trajectory Planning

Cubic trajectoryq(t)=a0+a1t+a2t2+a3t3q(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3q(t)=a0​+a1​t+a2​t2+a3​t3

Jerk(shows up on exams)jerk=q...=d3qdt3\mathrm{jerk} = \dddot{q} = \frac{d^3 q}{dt^3}jerk=q...​​=dt3d3q​

Point-to-point trajectory (minimal degree)(shows up on exams)q(t)=a1t+a0q(t) = a_1 t + a_0q(t)=a1​t+a0​

Velocity of a rotating body (exponential coordinates)(shows up on exams)v=−w^ rC=−(w×rC)v = -\hat{w}\, r_C = -(w \times r_C)v=−w^rC​=−(w×rC​)