Rigid Body Motion

Kinematics and Machine Dynamics · Chapter 5

Aykut C. Satici

Chapter overview

1. Rigid transformations and coordinate frames (§5.1)
2. Rotation matrices and their properties (§5.2.1)
3. Axis-angle, exponential coordinates, and Euler angles (§§5.2.2–5.2.3)
4. Homogeneous transformations (§5.3.1)
5. Twists and exponential rigid motion (§5.3.2)

Goal: express the position and orientation of a rigid body without ambiguity about frames or rotation order.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, Chapter 5, pp. 17–33. The chapter follows Murray, Li, and Sastry, A Mathematical Introduction to Robotic Manipulation.

The rigid-body idealization

A rigid body maintains the distance between every pair of material points:

\|p(t)-q(t)\|=\|p(0)-q(0)\|.

• A rigid motion is a continuous history of configurations.
• A rigid displacement relates an initial and a final configuration.
• A general displacement combines translation and rotation.

Deformation is neglected; individual points can still move relative to the fixed reference frame.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.1, pp. 17–19. Clarification: body-fixed coordinates are constant; separation vectors may rotate when expressed in a fixed frame.

Points, vectors, and frames

A point specifies a location. A free vector specifies a magnitude and direction.

v=q-p.

The same vector can connect different pairs of points.

A frame supplies an origin and three orthonormal axes. We use right-handed frames:

e_1\times e_2=e_3.

Coordinates have meaning only after a frame has been specified.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.1, pp. 17–19.

Distance preservation is not enough

The map (x,y,z)\mapsto(x,y,-z) preserves lengths but reflects the body.

A proper rigid transformation g must preserve both:

\|g(p)-g(q)\|=\|p-q\|, g_*(v\times w)=g_*(v)\times g_*(w),

where g_*(q-p)=g(q)-g(p) is the induced transformation of vectors.

The second condition preserves the handedness of coordinate frames.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.1, Definition 5.1, pp. 18–19.

Inner products and angles

Lengths determine inner products through the polarization identity:

v\cdot w=\frac14\left(\|v+w\|^2-\|v-w\|^2\right).

For the induced linear map on free vectors,

g_*(v)\cdot g_*(w)=v\cdot w.

Consequently, a rigid transformation preserves angles and orthogonality. A right-handed orthonormal body frame remains right-handed and orthonormal.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.1, Proposition 5.2, pp. 18–19. The induced map on free vectors is linear; translations cancel when subtracting points.

Orientation of a body frame

Compiled notes, Fig. 5.1; chapter based on Murray, Li, and Sastry.

Let A be the reference frame and B the body frame.

The columns of R_{ab} are the axes of B, expressed in A:

R_{ab}=\begin{bmatrix}x_{ab}&y_{ab}&z_{ab}\end{bmatrix}.

With coincident origins,

q_a=R_{ab}q_b.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.1, pp. 20–23.

Reading the frame subscripts

\boxed{v_a=R_{ab}v_b}

R_{ab} maps coordinates from B to A.

• The right subscript matches the input coordinates.
• The left subscript identifies the output coordinates.
• The physical vector is unchanged; its coordinate description changes.

The same matrix can describe an active rotation of a vector in a fixed frame. State which interpretation is being used.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.1, pp. 20–23.

The rotation group

Orthonormal columns imply

R^\mathsf{T}R=I,\qquad R^{-1}=R^\mathsf{T}.

A right-handed frame also requires \det R=+1.

\boxed{SO(3)=\{R\in\mathbb R^{3\times3}:R^\mathsf{T}R=I,\ \det R=1\}.}

An orthogonal matrix with determinant -1 includes a reflection.

More generally, SO(n) uses the same conditions for n\times n matrices.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.1, pp. 20–23.

Why rotations form a group

For R_1,R_2\in SO(3):

• Closure: (R_1R_2)^\mathsf{T}(R_1R_2)=I and \det(R_1R_2)=1.
• Identity: I leaves every vector unchanged.
• Inverse: R^{-1}=R^\mathsf{T}\in SO(3).
• Associativity: (R_1R_2)R_3=R_1(R_2R_3).

The configurations of a body rotating about a fixed point form SO(3). Its motion is a curve R(t) in that space.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.1, pp. 20–23.

Elementary rotations

With c=\cos\theta and s=\sin\theta,

R_x(\theta)=\begin{bmatrix}1&0&0\\0&c&-s\\0&s&c\end{bmatrix},\qquad R_y(\theta)=\begin{bmatrix}c&0&s\\0&1&0\\-s&0&c\end{bmatrix},

R_z(\theta)=\begin{bmatrix}c&-s&0\\s&c&0\\0&0&1\end{bmatrix}.

Positive angles follow the right-hand rule; angles in formulas are in radians.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.3, p. 26. Elementary matrices are introduced here before composition and reused for Euler angles.

Worked example: a quarter-turn

Rotate q=(2,1,0)^\mathsf{T} actively by 90^\circ about the fixed +z axis:

q'=R_z(\pi/2)q= \begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix} \begin{bmatrix}2\\1\\0\end{bmatrix} =\begin{bmatrix}-1\\2\\0\end{bmatrix}.

Checks: both vectors have length \sqrt5; applying R_z(-\pi/2) recovers q.

If R_{ab}=R_z(\pi/2) instead describes a frame orientation, the same multiplication converts q_b into q_a.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.1. Numerical example added for this deck; values are instructional, not textbook data.

Composing rotations

q_a=R_{ab}q_b,\qquad q_b=R_{bc}q_c

therefore

\boxed{R_{ac}=R_{ab}R_{bc}.}

Apply the rightmost transformation first.

For active rotations: a rotation about a fixed-frame axis premultiplies the current orientation; a rotation about a body-frame axis postmultiplies it.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.1, pp. 20–23.

Rotation order matters

Let e_z=(0,0,1)^\mathsf{T} and use active fixed-axis rotations.

R_z(\pi/2)R_x(\pi/2)e_z=e_x, R_x(\pi/2)R_z(\pi/2)e_z=-e_y.

Thus R_zR_x\ne R_xR_z in general.

Associativity lets us regroup products. It does not let us exchange their order.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.1, composition rule. Quarter-turn example added to make noncommutativity explicit.

The hat map and the cross product

For a=(a_1,a_2,a_3)^\mathsf{T}, define

\widehat a=\begin{bmatrix}0&-a_3&a_2\\a_3&0&-a_1\\-a_2&a_1&0\end{bmatrix}.

Then

\widehat a\,b=a\times b,\qquad \widehat a^\mathsf{T}=-\widehat a.

The vee map reverses the construction: (\widehat a)^\vee=a.

The set of 3\times3 skew-symmetric matrices is denoted \mathfrak{so}(3).

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §§5.2.1–5.2.2, pp. 22–24.

Rotations preserve cross products

For R\in SO(3),

R(v\times w)=(Rv)\times(Rw), \boxed{R\widehat wR^\mathsf{T}=\widehat{Rw}.}

Lengths are preserved because

\|R(q-p)\|^2=(q-p)^\mathsf{T}R^\mathsf{T}R(q-p)=\|q-p\|^2.

Together these properties establish that rotations are proper rigid transformations.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.1, pp. 20–23.

Rotation about an axis

Compiled notes, Fig. 5.2.

Let u be a unit vector along an axis through the origin, and let \theta be the rotation angle.

\frac{dq}{d\theta}=u\times q=\widehat u q.

The solution is

q(\theta)=e^{\widehat u\theta}q(0).

Hence R(u,\theta)=e^{\widehat u\theta}.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.2, pp. 23–24. The deck uses u for the unit rotation axis (omega in the source), reserving omega for angular velocity in Chapter 7.

The matrix exponential

e^{\widehat u\theta}=I+\theta\widehat u+ \frac{\theta^2}{2!}\widehat u^2+\frac{\theta^3}{3!}\widehat u^3+\cdots.

For a unit axis u,

\widehat u^2=uu^\mathsf{T}-I,\qquad \widehat u^3=-\widehat u.

Higher powers reduce to I, \widehat u, and \widehat u^2. Grouping odd and even powers produces the sine and cosine series.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.2, pp. 24–25.

Rodrigues’ formula

For \|u\|=1,

\boxed{R(u,\theta)=I+\widehat u\sin\theta+\widehat u^2(1-\cos\theta).}

For a general rotation vector \phi, put \vartheta=\|\phi\|:

e^{\widehat\phi}=I+\frac{\sin\vartheta}{\vartheta}\widehat\phi +\frac{1-\cos\vartheta}{\vartheta^2}\widehat\phi^2.

At \vartheta=0, take the continuous limit: e^0=I. Near zero, use series or numerically stable implementations.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.2, p. 25. Uses the vector norm in the general formula and explicitly includes the zero-angle limit.

Why the exponential is a rotation

Since \widehat u^\mathsf{T}=-\widehat u,

(e^{\widehat u\theta})^\mathsf{T}e^{\widehat u\theta} =e^{-\widehat u\theta}e^{\widehat u\theta}=I.

The determinant is +1: it varies continuously from \det I=1 and an orthogonal matrix’s determinant is always \pm1.

Every rotation in SO(3) has an axis-angle representation, though that representation is not unique.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.2, Propositions 5.6–5.7, pp. 25–26.

Recovering axis and angle

For a valid R\in SO(3), choose the principal angle 0\le\theta\le\pi:

\theta=\cos^{-1}\!\left(\frac{\operatorname{tr}R-1}{2}\right).

When 0<\theta<\pi,

u=\frac{1}{2\sin\theta} \begin{bmatrix}r_{32}-r_{23}\\r_{13}-r_{31}\\r_{21}-r_{12}\end{bmatrix}.

In floating-point arithmetic, clip the inverse-cosine argument to [-1,1] after checking that R is a valid rotation.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.2, Eq. (5.13), p. 26. Corrects the missing subscript in r31 and restricts the axis formula to 0 < theta < pi.

Axis-angle special cases

• \theta=0: R=I; the axis is arbitrary.
• \theta=\pi: the formula dividing by \sin\theta is undefined. Find a unit eigenvector satisfying Ru=u, or use R+I=2uu^\mathsf{T}.
• Nonuniqueness: (u,\theta) and (-u,-\theta) produce the same rotation; adding 2\pi also preserves it.

Example: R=\operatorname{diag}(1,-1,-1) is a half-turn about e_x.

The rotation vector is \phi=u\theta; it is a coordinate representation, not a globally unique label.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.2, p. 26. The pi case repairs an omission in the compiled notes; diagonal example is an instructional addition.

ZYZ Euler angles

Starting with coincident frames, rotate about the moving axes:

1. z by \alpha;
2. the new y by \beta;
3. the new z by \gamma.

\boxed{R=R_z(\alpha)R_y(\beta)R_z(\gamma).}

The axes and their order are part of the definition. Three numbers without a convention do not specify an orientation.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.3, pp. 26–27.

ZYZ matrix and angle recovery

Write c_\alpha=\cos\alpha, s_\alpha=\sin\alpha, and similarly for the other angles.

R=\begin{bmatrix} c_\alpha c_\beta c_\gamma-s_\alpha s_\gamma&-c_\alpha c_\beta s_\gamma-s_\alpha c_\gamma&c_\alpha s_\beta\\ s_\alpha c_\beta c_\gamma+c_\alpha s_\gamma&-s_\alpha c_\beta s_\gamma+c_\alpha c_\gamma&s_\alpha s_\beta\\ -s_\beta c_\gamma&s_\beta s_\gamma&c_\beta \end{bmatrix}.

For the branch 0<\beta<\pi,

\begin{aligned} \beta&=\operatorname{atan2}(\sqrt{r_{31}^2+r_{32}^2},r_{33}),\\ \alpha&=\operatorname{atan2}(r_{23},r_{13}),\qquad \gamma=\operatorname{atan2}(r_{32},-r_{31}). \end{aligned}

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.3, Eq. (5.14) and following formulas, p. 27. atan2 is simplified on the stated positive-sine beta branch.

Euler singularities and other conventions

At \beta=0,

R_z(\alpha)R_y(0)R_z(\gamma)=R_z(\alpha+\gamma).

Only the sum is determined: \alpha and \gamma cannot be recovered separately. ZYZ also has a singularity at \beta=\pi.

A common yaw–pitch–roll convention is

R=R_z(\psi)R_y(\vartheta)R_x(\varphi).

This is intrinsic ZYX, equivalently fixed-axis roll, then pitch, then yaw. Do not mix it with intrinsic XYZ.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.2.3, p. 27. Explicitly resolves the source wording about body versus fixed axes; singularity explanation is added for completeness.

Position and orientation together

Compiled notes, Fig. 5.3.

Let p_{ab} locate the origin of B in A.

\boxed{q_a=p_{ab}+R_{ab}q_b.}

A configuration is described by translation and orientation:

SE(3)\cong\mathbb R^3\times SO(3)

as a set of configurations.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3, pp. 27–29. The group product is a semidirect product, not componentwise addition/multiplication; the displayed identification is explicitly at the set level.

Homogeneous coordinates

Represent points and free vectors differently:

\bar q=\begin{bmatrix}q\\1\end{bmatrix},\qquad \bar v=\begin{bmatrix}v\\0\end{bmatrix}.

The affine map becomes a matrix multiplication:

g_{ab}=\begin{bmatrix}R_{ab}&p_{ab}\\0&1\end{bmatrix},\qquad \bar q_a=g_{ab}\bar q_b.

For a vector, g_{ab}\bar v_b=(R_{ab}v_b,0)^\mathsf{T}: translation cancels.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3.1, pp. 29–30.

Composing and inverting rigid transforms

\boxed{g_{ac}=g_{ab}g_{bc} =\begin{bmatrix}R_{ab}R_{bc}&p_{ab}+R_{ab}p_{bc}\\0&1\end{bmatrix}.}

Rotate p_{bc} into frame A before adding it to p_{ab}.

\boxed{g_{ab}^{-1}=g_{ba} =\begin{bmatrix}R_{ab}^\mathsf{T}&-R_{ab}^\mathsf{T}p_{ab}\\0&1\end{bmatrix}.}

Closure, identity, inverses, and associativity make these matrices the group SE(3).

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3.1, pp. 29–30.

Worked example: a transformed point

Let R_{ab}=R_z(\pi/2), p_{ab}=(1,2,0)^\mathsf{T}, and q_b=(2,0,0)^\mathsf{T} m.

q_a=\begin{bmatrix}1\\2\\0\end{bmatrix} +\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix} \begin{bmatrix}2\\0\\0\end{bmatrix} =\begin{bmatrix}1\\4\\0\end{bmatrix}\ \mathrm m.

The inverse check is q_b=R_{ab}^\mathsf{T}(q_a-p_{ab}).

For the free vector v_b=(2,0,0)^\mathsf{T} m, v_a=(0,2,0)^\mathsf{T} m. Do not add p_{ab} to it.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3.1. Numerical example added for this deck.

Rotation about an offset axis

Compiled notes, Fig. 5.4(a); axis direction \omega in the figure is u here.

For unit axis u through a fixed point q,

\frac{dp}{d\theta}=u\times(p-q) =\widehat u p+v, v=-u\times q.

This motion includes a translation when the axis does not pass through the coordinate origin.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3.2, pp. 31–32. Corrects the source homogeneous-matrix typo: the constant column is -u cross q, not -u cross p.

Twist coordinates

Use the linear-first ordering of the textbook:

\xi=\begin{bmatrix}v\\u\end{bmatrix}\in\mathbb R^6,\qquad \widehat\xi=\begin{bmatrix}\widehat u&v\\0&0\end{bmatrix}\in\mathfrak{se}(3).

The wedge map builds \widehat\xi from \xi; the vee map reverses it.

\frac{d\bar p}{d\theta}=\widehat\xi\bar p \quad\Longrightarrow\quad \bar p(\theta)=e^{\widehat\xi\theta}\bar p(0).

A twist is a generator of rigid motion. For a revolute axis, v is generally not the velocity of the selected body origin.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3.2, pp. 31–33. Here the normalized generator is multiplied by joint displacement; actual velocity additionally includes the joint rate.

Prismatic motion

Compiled notes, Fig. 5.4(b).

For a unit translation direction v, use

\widehat\xi=\begin{bmatrix}0&v\\0&0\end{bmatrix}.

Since \widehat\xi^2=0,

e^{\widehat\xi s}=\begin{bmatrix}I&sv\\0&1\end{bmatrix}.

Here s is a length, not an angle.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3.2, pp. 32–33.

Computing a twist exponential

For \|u\|=1 and \widehat\xi=\begin{bmatrix}\widehat u&v\\0&0\end{bmatrix},

e^{\widehat\xi\theta}=\begin{bmatrix}R(\theta)&V(\theta)v\\0&1\end{bmatrix}, V(\theta)=I\theta+(1-\cos\theta)\widehat u +(\theta-\sin\theta)\widehat u^2.

For a pure revolute joint through q, v=-u\times q, so

V(\theta)v=(I-R(\theta))q.

The closed form follows by integrating R(\tau)v from 0 to \theta.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3.2. Closed-form evaluation is an instructional extension of the source exponential definition; it is derived by integrating the rotation exponential.

Worked example: an offset revolute joint

Let u=e_z, let the axis pass through q=(1,0,0)^\mathsf{T} m, and rotate by \pi/2.

v=-e_z\times q=(0,-1,0)^\mathsf{T}\ \mathrm m, p_{\rm trans}=(I-R_z(\pi/2))q=(1,-1,0)^\mathsf{T}\ \mathrm m.

For an initial point p(0)=(2,0,0)^\mathsf{T} m,

p(\pi/2)=R_z(\pi/2)p(0)+p_{\rm trans}=(1,1,0)^\mathsf{T}\ \mathrm m.

Its distance from the axis remains 1 m.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3.2. Numerical example added to check the offset-axis sign and finite transformation.

Displacement versus frame conversion

A frame transformation relates two coordinate descriptions:

\bar q_a=g_{ab}\bar q_b.

An exponential displacement moves points while keeping one reference frame:

\bar p(\theta)=e^{\widehat\xi\theta}\bar p(0).

For a generator expressed in the fixed frame,

\boxed{g_{ab}(\theta)=e^{\widehat\xi\theta}g_{ab}(0).}

The initial pose need not be the identity. Every proper rigid displacement can be represented by a twist exponential.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, §5.3.2, Eq. (5.27) and final proposition, p. 33.

Chapter summary and references

• SO(3) represents orientation; SE(3) represents position and orientation.
• Frame subscripts determine valid matrix products.
• Axis-angle and Euler coordinates need special-case handling.
• Homogeneous coordinates distinguish points from vectors.
• Twist exponentials describe joint displacements.

Reading: compiled notes, Chapter 5, pp. 17–33; Murray, Li, and Sastry, A Mathematical Introduction to Robotic Manipulation, Chapter 2 (author’s book page).

Next: Chapter 6 — Position-Level Analysis.

Source: Satici, Kinematics and Machine Dynamics, Fall 2020 compiled notes, Chapter 5. The original bibliography lists a 2017 edition under Murray; the author page confirms the complete authorship and 1994 first edition.