I. Orientation: Mathematical Description of Robotic Geometry

Robotics represents the position, orientation, motion, and physical properties of bodies using coordinate systems, vectors, calculus, and matrices. A robot configuration is usually described relative to a fixed reference frame, while transformations connect coordinates measured in different frames. Differential quantities describe infinitesimal geometry, and integrals accumulate these quantities along lines, surfaces, or volumes.

• Reference frame: A frame consists of an origin and ordered basis directions, such as (x), (y), and (z).
• Right-handed convention: Positive rotation follows the right-hand rule; (\hat{\mathbf{x}}\times\hat{\mathbf{y}}=\hat{\mathbf{z}}).
• Coordinate representation: A point is represented by coordinates, while a vector represents displacement, velocity, force, or direction.
• Units and dimensions: Length may be measured in metres, area in (\text{m}^2), volume in (\text{m}^3), and angles in radians.
• Linear algebra principle: Matrices encode linear mappings, coordinate changes, rotations, scaling, and system equations.
• Robotics convention: A rigid-body pose combines translation and orientation; homogeneous transformation matrices represent both in one matrix.

II. Introduction to Coordinate systems

A coordinate system assigns numerical coordinates to geometric points relative to an origin and a set of basis vectors. In robotics, the choice of coordinates determines how position, orientation, and motion are calculated.

A. Introduction to Coordinate systems

The main purpose of a coordinate system is to convert geometric information into algebraic form suitable for computation.

• Cartesian coordinates: In three dimensions, a point (P) is written as
  [
  \mathbf{p}=
  \begin{bmatrix}
  x\y\z
  \end{bmatrix},
  ]
  where (x,y,z) are signed distances along mutually perpendicular axes.
• Basis vectors: The position vector can be expanded as
  [
  \mathbf{p}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}},
  ]
  where (\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}) are unit basis vectors.
• Coordinate frames: A frame ({A}) is commonly specified by origin (O_A) and basis ((\hat{\mathbf{x}}_A,\hat{\mathbf{y}}_A,\hat{\mathbf{z}}_A)). The same physical point has different numerical coordinates in different frames.
• Polar and cylindrical coordinates: In two dimensions,
  [
  x=r\cos\theta,\qquad y=r\sin\theta.
  ]
  In cylindrical coordinates, ((r,\theta,z)) describes radius, angle, and height.
• Spherical coordinates: A point may be described by ((\rho,\theta,\phi)), where (\rho) is distance from the origin, (\theta) is azimuth, and (\phi) is the polar angle.
• Orientation: A rotation matrix (R) maps coordinates from one orthonormal frame to another. Its columns are the new frame’s unit axes expressed in the old frame.
• Frame transformation: If (^{A}\mathbf{p}) and (^{B}\mathbf{p}) denote the same point in frames (A) and (B), then a rigid transformation has the form
  [
  {}^{A}\mathbf{p}= {}^{A}R{B}\,{}^{B}\mathbf{p}+{}^{A}\mathbf{t}{B},
  ]
  where (R) is orientation and (\mathbf{t}) is translation.

III. Differential Length, Area and Volume

Differential measures describe infinitesimally small geometric elements. They are essential for calculating mass, centre of mass, force distributions, and geometric quantities over robot links and workspaces.

A. Differential Length, Area and Volume

The differential element depends on the coordinate system and must include the appropriate scale factors.

• Cartesian length: For a displacement ((dx,dy,dz)),
  [
  ds=\sqrt{dx^2+dy^2+dz^2}.
  ]
  Here (ds) is an infinitesimal length.
• Cartesian area: A rectangular differential surface in the (xy)-plane is
  [
  dA=dx\,dy.
  ]
  The differential area vector is (d\mathbf{A}=\hat{\mathbf{n}}\,dA), where (\hat{\mathbf{n}}) is the unit normal.
• Cartesian volume: A small rectangular volume is
  [
  dV=dx\,dy\,dz.
  ]
  For constant density (\rho), a small mass element is (dm=\rho\,dV).
• Polar coordinates: The differential area is
  [
  dA=r\,dr\,d\theta.
  ]
  The factor (r) accounts for the fact that an angular interval represents a longer arc farther from the origin.
• Cylindrical coordinates: The differential volume is
  [
  dV=r\,dr\,d\theta\,dz.
  ]
  Here (r) is radial distance, (\theta) azimuth, and (z) height.
• Spherical coordinates: The differential volume is
  [
  dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta.
  ]
  The factors (\rho^2) and (\sin\phi) arise from radial and angular stretching.
• Metric interpretation: A general curvilinear system with scale factors (h_1,h_2,h_3) has
  [
  ds^2=h_1^2du_1^2+h_2^2du_2^2+h_3^2du_3^2,\qquad
  dV=h_1h_2h_3\,du_1du_2du_3.
  ]
  The scale factors convert coordinate changes into physical distances.

B. Applications and limitations

Differential elements connect local geometry with global physical properties.

• Mass calculation: For density (\rho(\mathbf{r})),
  [
  m=\int_V \rho(\mathbf{r})\,dV.
  ]
  If density is uniform, (m=\rho V).
• Centre of mass: Its position is
  [
  \mathbf{r}_{c}=\frac{1}{m}\int_V \mathbf{r}\,\rho(\mathbf{r})\,dV.
  ]
• Coordinate dependence: Using (dx\,dy\,dz) in cylindrical coordinates would omit the required factor (r) and produce an incorrect result.
• Local approximation: Differential elements are treated as sufficiently small that curvature and variation are negligible within each element.

IV. Line, Surface and Volume Integrals

Integrals accumulate a scalar or vector quantity over a geometric domain. In robotics, they describe work, flux, mass, inertia, and distributed loads.

A. Line, Surface and Volume Integrals

A line integral follows a curve, a surface integral covers a two-dimensional boundary, and a volume integral covers a three-dimensional region.

• Line integral of a scalar: For scalar field (f(\mathbf{r})) along curve (C),
  [
  \int_C f\,ds.
  ]
  If (\mathbf{r}(t)), (a\le t\le b), parameterizes the curve, then
  [
  ds=|\mathbf{r}'(t)|dt.
  ]
• Line integral of a vector field: For force (\mathbf{F}) acting along displacement (d\mathbf{r}),
  [
  W=\int_C\mathbf{F}\cdot d\mathbf{r}.
  ]
  The result (W) is mechanical work, measured in joules.
• Surface integral of a scalar: For surface (S),
  [
  \int_S g\,dA
  ]
  accumulates a surface density (g), such as surface mass density.
• Flux integral: The flux of vector field (\mathbf{F}) through oriented surface (S) is
  [
  \iint_S\mathbf{F}\cdot\hat{\mathbf{n}}\,dA.
  ]
  The dot product selects the component normal to the surface.
• Volume integral: For a scalar field (h) over volume (V),
  [
  \iiint_V h\,dV.
  ]
  For example, total mass is (\iiint_V\rho\,dV).
• Orientation: Reversing a curve reverses a vector line integral. Reversing a surface normal changes the sign of flux.

• Conservation connection: The divergence theorem relates outward flux to volume accumulation:
  [
  \iint_{\partial V}\mathbf{F}\cdot\hat{\mathbf{n}}\,dA

  \iiint_V\nabla\cdot\mathbf{F}\,dV.
  ]
  Here (\partial V) is the closed boundary of (V), and (\nabla\cdot\mathbf{F}) is divergence.

B. Worked application

A uniform rod of length (L), linear density (\lambda), and position (\mathbf{r}(s)) has mass

[
m=\int_0^L\lambda\,ds=\lambda L.
]

Its centre of mass is

[
\mathbf{r}_c=\frac{1}{m}\int_0^L\mathbf{r}(s)\lambda\,ds.
]

For a straight rod from (\mathbf{r}_0) to (\mathbf{r}_1), this gives (\mathbf{r}_c=(\mathbf{r}_0+\mathbf{r}_1)/2).

V. Matrix and its decompositions

A matrix is a rectangular array representing numbers, vectors, or a mapping between vector spaces. Matrix decompositions rewrite a matrix as a product of structured matrices, making robotic computation more efficient and numerically reliable.

A. Matrix and its decompositions

Matrices support coordinate transformations, kinematic equations, least-squares estimation, and dynamic models.

• Matrix notation: An (m\times n) matrix (A=[a{ij}]) has (m) rows and (n) columns. Its element (a{ij}) lies in row (i), column (j).
• Matrix-vector mapping: For (A\in\mathbb{R}^{m\times n}) and (\mathbf{x}\in\mathbb{R}^n),
  [
  \mathbf{y}=A\mathbf{x},\qquad
  yi=\sum{j=1}^{n}a_{ij}x_j.
  ]
• Identity and inverse: The identity matrix (I) satisfies (AI=A). If (A^{-1}) exists,
  [
  AA^{-1}=A^{-1}A=I.
  ]
  A square matrix is invertible only when (\det(A)\ne0).
• Transpose: (A^T) interchanges rows and columns. An orthogonal matrix satisfies
  [
  Q^TQ=I,\qquad Q^{-1}=Q^T.
  ]
• Rotation matrix: A proper three-dimensional rotation satisfies
  [
  R^TR=I,\qquad \det(R)=1.
  ]
  These conditions preserve lengths, angles, and orientation.
• Homogeneous transformation: A rigid pose is represented by
  [
  T=
  \begin{bmatrix}
  R&\mathbf{t}\
  \mathbf{0}^T&1
  \end{bmatrix},
  ]
  where (R) is (3\times3), (\mathbf{t}) is a (3\times1) translation, and (\mathbf{0}^T=[0\ 0\ 0]).

B. Matrix and its decompositions

Decomposition separates a matrix into factors with useful algebraic properties.

• LU decomposition: A square matrix may be factored as
  [
  A=LU,
  ]
  where (L) is lower triangular and (U) is upper triangular. With row exchanges, the practical form is (PA=LU), where (P) is a permutation matrix.
• QR decomposition: A matrix can be written
  [
  A=QR,
  ]
  where (Q^TQ=I) and (R) is upper triangular. QR is useful for least-squares problems because orthogonal transformations preserve numerical length.
• Cholesky decomposition: If (A) is symmetric positive definite, then
  [
  A=LL^T,
  ]
  where (L) is lower triangular with positive diagonal entries. Robot inertia and covariance matrices often have properties suited to this factorization.
• Eigenvalue decomposition: For suitable square (A),
  [
  A=V\Lambda V^{-1},
  ]
  where columns of (V) are eigenvectors and (\Lambda) contains eigenvalues satisfying
  [
  A\mathbf{v}=\lambda\mathbf{v}.
  ]
• Singular value decomposition: Every real matrix has
  [
  A=U\Sigma V^T,
  ]
  where (U) and (V) are orthogonal and (\Sigma) contains nonnegative singular values. SVD is valuable for rank analysis, pseudoinverses, and redundant robot control.
• Pseudoinverse: For an underdetermined or overdetermined system, the Moore–Penrose pseudoinverse (A^+) gives a least-squares or minimum-norm solution:
  [
  \mathbf{x}=A^+\mathbf{b}.
  ]
  When (A=U\Sigma V^T), (A^+=V\Sigma^+U^T), where (\Sigma^+) replaces each nonzero singular value (\sigma) by (1/\sigma).

C. Applications and limitations

Choosing a decomposition depends on matrix structure, purpose, and numerical conditioning.

• Solving systems: LU solves (A\mathbf{x}=\mathbf{b}) through triangular solves rather than explicitly forming (A^{-1}).
• Least squares: QR solves
  [
  \min_{\mathbf{x}}|A\mathbf{x}-\mathbf{b}|_2
  ]
  without directly forming the potentially unstable matrix (A^TA).
• Singularity detection: A very small singular value indicates a direction in which the mapping loses sensitivity; robot Jacobians near such configurations cause poor motion control.
• Computational efficiency: Triangular matrices require fewer operations than dense matrices, while sparse or structured matrices can reduce memory use.
• Numerical limitation: Explicit inversion can amplify round-off error. Factorizations are generally preferred for solving equations, especially when coordinates or sensor data are ill-conditioned.