# slama.dev

## Robotics 1

notes , released 30. 11. 2022, updated 16. 12. 2022; available in [PDF]

### Preface

This website contains my lecture notes from a lecture by Lorenzo Masia from the academic year 2022/2023 (University of Heidelberg). If you find something incorrect/unclear, or would like to contribute, feel free to submit a pull request (or let me know via email).

Note that since the course hasn’t ended yet, the notes are not complete and won’t be until the end of the semester.

### Robotics Components

Definition (link/member): individual bodies making up a mechanism

Definition (joint): connection between multiple links

Definition (kinematic pair): two links in contact such that it limits their relative

• low-order kinematic pairs: point of contact is a surface (eg. revolute/prismatic/spherical)
• high-order kinematic pairs: point of contact is a dot/line (eg. gears)

Definition (kinematic chain): assembly of links connected by joints

• is a closed loop (left image) if every link is connected to every other by at least two paths (including the ground – imagine it as a single link), else it’s open (right image)

#### Degrees of Freedom

Definition (deegree of freedom) of a mechanical system is the number of independent parameters that define its configuration. It can be calculated by the Grübler Formula:

$\text{DOF} = \lambda (n - 1) - \sum_{i = 1}^{j} c_i$

where

• $\lambda \ldots$ DOF of the operating space (3 for 2D, 6 for 3D)
• in 2D it’s 2 for orientation and 1 for position
• in 3D it’s 3 for orientation and 3 for position
• $n \ldots$ number of links
• $j \ldots$ number of kinematic pairs
• $c_j \ldots$ degree of constraint of the $i$-th kinematic pair
• eg. a revolute joint in 2D takes away 2 DOF (we can only rotate)

For example, the following has $3 (4 - 1) - 2 \cdot 4 = 1$ DOF:

Definition (coincident joints): when there are more than two kinematic pairs in the same joint

#### Actuators

Definition (actuator): a mechanical device for moving or controlling something

• DC brushed motor: based on Lorentz’ force law (electromagnetic fields)
• brushed because the metal brush powers the magnets (they’re turning)
• brushless uses the position of the motor to turn on/off currents for specific windings
• usually contains gears reductions to trade torque for speed and sensors to measure the position of the motor (see further)

• since the voltage controls the motor but setting it to a specific value is impractical, pulse width modulation (PWM) is used:

• to measure the position of the motor, optical shaft encoders (incremental/absolute) are used:

### Kinematics

• establishment of various coordinate systems to represent the positions and orientations of rigid objects and with transformations among these coordinate systems

• the dimension of the configuration space ($n$) must be larger or equal to the dimension of the task space ($m$) to ensure the existence of kinematic solutions

#### Forward (direct) Kinematics

Definition (forward/direct kinematics): the process of finding the position/orientation of the end-effector $(r_1, \ldots, r_m)$ given a set of joint parameters $(q_1, \ldots, q_n)$.

$(r_1, \ldots, r_m) = F(q_1, \ldots, q_n)$

##### Body Pose

The pose/frame of a rigid body can be described by its position and orientation (wrt. a reference frame).

• the position is a vector $P \in \mathbb{R}^3$
• the rotation is an orthonormal matrix $R \in \mathbb{R}^{3 \times 3}$ with $\det(R) = 1$ (a determinant of $-1$ would flip the object, we only want rotation)
• due to orthogonality: $R^T = R^{-1}$

$R = \begin{bmatrix} x' & y' & z' \end{bmatrix} = \begin{bmatrix} x_x' & y_x' & z_x' \\ x_y' & y_y' & z_y' \\ x_z' & y_z' & z_z' \end{bmatrix}$

The elementary rotations about each of the axes are the following:

\begin{aligned} R_x(\alpha) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \alpha & -\sin \alpha \\ 0 & \sin \alpha & \cos \alpha \end{bmatrix} \\ R_y(\alpha) &= \begin{bmatrix} \cos \alpha & 0 & \sin \alpha \\ 0 & 1 & 0 \\ -\sin \alpha & 0 & \cos \alpha \end{bmatrix} \\ R_z(\alpha) &= \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ \end{aligned}

• important to remember for calculating robot kinematics (or just think about what the rotation about an axis is doing to the other two axes, it’s not too hard to remember)
• positive values for the rotation are always counter-clockwise

When discussing multiple frames, we use the following notation:

${\scriptstyle \text{to} \atop \scriptstyle\text{from}} R = {\scriptstyle \text{to} \atop \scriptstyle\text{}} R {\scriptstyle \text{} \atop \scriptstyle\text{from}} = R {\scriptstyle \text{to} \atop \scriptstyle\text{from}}$

E.g. if we have the same point $p_0, p_1, p_2$ in three different frames, we know that

\begin{aligned} p_1 &= R^1_2 \cdot p_2 \\ p_0 &= R^0_1 \cdot p_1 \\ p_0 &= R^0_2 \cdot p_2 \end{aligned}

For a minimal representation, we will use Euler’s ZYZ angles, which does three rotations:

\begin{aligned} R &= R_z (\varphi) R_{y'} (\vartheta) R_{z''} (\psi) \\ &= \begin{bmatrix} c_{\varphi} c_{\vartheta} c_{\psi} - s_{\varphi} s_{\psi} & -c_{\varphi} c_{\vartheta} s_{\psi} - s_{\varphi} c_{\psi} & c_{\varphi} s_{\vartheta} \\ s_{\varphi} c_{\vartheta} c_{\psi} + c_{\varphi} s_{\psi} & -s_{\varphi} c_{\vartheta} s_{\psi} + c_{\varphi} c_{\psi} & s_{\varphi} s_{\vartheta} \\ -s_{\vartheta} c_{\psi} & s_{\vartheta} s_{\psi} & c_{\vartheta} \end{bmatrix} \end{aligned}

For the inverse problem (calculating angles from a matrix of numbers), we can do \begin{aligned} \varphi &= \mathrm{atan}2 (r_{2,3}, r_{1, 3}) \\ \vartheta &= \mathrm{atan}2 \left(\sqrt{r_{1,3}^2 + r_{2,3}^2}, r_{3, 3}\right) \quad \vartheta \in (0, \pi)\ \text{since we took + sign} \\ \psi &= \mathrm{atan}2 \left(r_{3,2}, -r_{3,1}\right) \end{aligned}

• if we divide by zero somewhere we get degenerate solutions where we can only get the sum of the angles (one of the problems with Euler angles)
• alternative is RPY angles, which are also three rotations (roll, pitch, yaw) and are just as bad
##### Denavit & Hartenberg Notation

For relating the base and the end effector, we need to both rotate and translate, so we’ll use homogeneus coordinates (matrix is $4 \times 4$, encoding both rotation and translation):

${\scriptstyle \text{A} \atop \scriptstyle\text{EE}} A = \begin{bmatrix} {\scriptstyle \text{A} \atop \scriptstyle\text{EE}} R & {\scriptstyle \text{A} \atop \scriptstyle\text{EE}} P \\ 000 & 1\end{bmatrix}$

To create the homogeneous coordinates in a standardized way, we use the Denavit & Hartenberg (DH) notation which systematically relates the frames of two consecutive links. We have 4 parameters, each of which relates frame $i$ to frame $i - 1$:

parameter meaning
link length $a_i$ distance between $z_i$ and $z_{i - 1}$ along $x_i$
link offset $d_i$ distance between $x_i$ and $x_{i - 1}$ along $z_i$
link twist $\alpha_i$ angle between $z_i$ and $z_{i - 1}$ around $x_i$
joint angle $\theta_i$ angle between $x_i$ and $x_{i - 1}$ around $z_i$

Example: anthropomorphic arm (3 revolute joins):

$a_i$ $d_i$ $\alpha_i$ $\theta_i$
1 $0$ $\pi / 2$ $0$ $\theta_1^*$
2 $a_2$ $0$ $0$ $\theta_2^*$
3 $a_3$ $0$ $0$ $\theta_3^*$

We then get the following transformations:

\begin{aligned} A^0_1 (\theta_1) &= \begin{bmatrix} c_1 & 0 & s_1 & 0 \\ s_1 & 0 & -c_1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \quad R = R_z(\theta_1) \cdot R_x(\pi / 2) \\ A^{i-1}_i (\theta_1) &= \begin{bmatrix} c_i & -s_i & 0 & a_i c_i \\ s_i & c_i & 0 & a_i s_i \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \quad i = 2, 3\\ \end{aligned}

To get the final transformations, we can multiply the matrices and get $T^0_3 (\mathbf{q}) = A^0_1 A^1_2 A^2_3$

#### Inverse Kinematics

Definition (inverse kinematics): the process of finding a set of joint parameters $(q_1, \ldots, q_n)$ given the position/orientation of the end-effector $(r_1, \ldots, r_m)$.

$(q_1, \ldots, q_n) = G(r_1, \ldots, r_m)$

Definition (primary workspace): set $\mathrm{WS}_1$ of all positions $p$ that can be reached with at least one orientation $R$

Definition (secondary workspace): set $\mathrm{WS}_2$ of all positions $p$ that can be reached with any orientation $R$

##### Analytical solution (closed form)
• preferred (if it can be found)
• use geometric inspection, solve system of equations

Example: spherical wrist (3 revolute joins):

$T_6^3(\mathbf{q}) = \begin{bmatrix} c_4 c_5 c_6 - s_4 s_6 & -c_4 c_5 s_6 - s_4 c_6 & c_4 s_5 & c_4 s_5 d_6 \\ s_4 c_5 c_6 + c_4 s_6 & -s_4 c_5 s_6 + c_4 c_6 & s_4 s_5 & s_4 s_5 d_6 \\ -s_5 c_6 & s_5 s_6 & c_5 & c_5 d_6 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

The matrix $R$ is a ZYZ Euler rotation matrix and can be solved as such (see Body pose).

• we use indexes $3$ to $6$, because the wrist is usually at an end of another manipulator

Definition (decoupling): dividing inverse kinematics problem for two simpler problems, inverse position kinematics and inverse orientation kinematics

• is applicable for manipulators with at least 6 joints where the last 3 intersect at a point

The general approach is the following:

1. calculate the orientations and position where the wrist needs to be
2. calculate the orientations of the rest of the robot
##### Numerical solution (iterative form)

TBA

• Gradient method derivation (might be on the exam)
# will be typeset shortly
H(q) = 1/2 ||r_d f_r(q)||^2
dH / dq = 1/2  2 (-1) [r_d - f_r(q)]^T J_r(q)
dH / dq = (-1) [r_d - f_r(q)]^T J_r(q)  // nabla transposes
nabla q H(q) = (-1) [r_d - f_r(q)] J_r(q)^T  // nabla transposes