Assistive Robotics

HOMOGENEOUS TRANSFORMATIONS

On the same time, I'm rotating and translating. p is the representation of p respect to the frame A and p is the representation of point p respect to the frame B. p is the distance of the centre of the reference frame B w.r.t the centre of the reference frame A considering x y z of the reference frame A. Letter in the top left means that we are dealing about the representation of something with respect to that frame. Bottom right the original frame.

We can have the coordinate of the point P w.r.t. the RF A knowing:

1. the distance between the 2 centres O and O defined over the frame A, A B
2. the rotation of the frame B w.r.t. the frame A (rotation matrix define how B is rotated w.r.t. A)
3. the position of the point P w.r.t. the reference frame B

So, we know the blue and brown vector, and the rotation matrix on how B is rotating w.r.t. A.

Is possible to obtain the red vector (p) summing the distance between the 2 centres of the reference frames A and B and the product of the rotation matrix R and the vector p.

1. Transforms a vector from one frame to another.

2. Transforms the representation of a position vector (applied vector starting from the origin of the frame) from a given frame to another frame.

3. It is a roto-translation operator on vectors in the three-dimensional space.

4. It is always invertible (T^-1 = T). If we have the pose of B with respect to A, the inverse gives us the pose of A with respect to B.

5. Can be composed, i.e., T = T₁ T₂ (it does not commute!). It is a very important property because if we have many reference frames, by the composition of all successive transformation matrices we can obtain the pose of the last reference frame with respect to the first one.

11. DIRECT AND INVERSE KINEMATICS OF THE MANIPULATOR

DEFINING A ROBOT TASK

RF is the world reference frame and can be defined in a certain part of the human space.

RF is the base reference frame, this is always defined at the base of the robot.

Also, the end effector will always have a frame on it. These 2 frames are very important because the

Il tuo obiettivo è trovare la relazione tra il frame dell'end effector e il frame della base del robot. RF è fisso mentre il frame dell'end effector si muoverà e ruoterà a seconda di come si muove il robot. Dobbiamo definire l'operazione di roto-traslazione tra questi 2 frame (blu e giallo). W- T è sempre fisso perché si definisce a priori RF e RFB W BB- T dipende dalle variabili congiunte. Le variabili congiunte saranno chiamate q. Una variabile congiunta è una rotazione (se hai un giunto rotativo o rotativo) o è uno spostamento di traslazione (se hai un giunto prismático). Quindi, q sarà il valore di rotazione o traslazione di quel particolare giunto. La variabile congiunta sarà 6, una per ogni giunto.

CINEMATICA: FORMULAZIONE E PARAMETRIZZAZIONE

La cinematica diretta significa che conosco le variabili congiunte e voglio trovare la posizione e l'orientamento nello spazio cartesiano. La cinematica diretta ti permette di passare dallo spazio congiunto allo spazio cartesiano. Una volta che conosci il valore delle

Joint variables provide a Cartesian formulation where the end effector's position and orientation are considered. Direct kinematics allows us to obtain the position and orientation of the end effector by knowing only the value of the joint variables, taking into account the geometry of the robot.

Inverse kinematics, on the other hand, means that we know the position and orientation in the Cartesian space and we want to find the joint variables that allow us to reach this particular point and orientation.

The choice of parameterization q (joints position) is an unambiguous and minimal characterization of the robot configuration. It represents the degrees of freedom (dof) or the positions of the robot joints (rotational or translational).

The choice of parameterization r is usually the end-effector position and orientation. It provides a compact description of the position and/or orientation (pose) variables of interest for the required task. Typically, it includes m n and m 6, but none of these is strictly necessary.

The direct kinematics is the minimal characterization of the robot configuration.

In the joint space, the number n of joint variables is equal to the number of joints. While in the Cartesian space, the number m of variables (typically 6 in space and 3 in plane but sometimes also 4) represents the position and orientation of the end effector frame.

DIRECT AND INVERSE KINEMATICS OF THE MANIPULATOR

DIRECT KINEMATICS PROBLEM

Given the values of the joint parameters, find the end-effector pose.

It is an analytical problem, a problem with always a solution where input data are in the form r = f(q) with q = (q1, ..., qn). One input, one output. Given the value of q, 1, ..., n until the end of your joints, r (position and orientation of end effector) will be a function of this q and you have only one solution.

By knowing the value of q, we want to know the position and orientation of the end effector. The structure of the direct kinematics function depends on the chosen r: r = f(q) that could be computed with two methods (examples on notebook):

1. geometric/by inspection (oral question)
2. ...

systematic: assigning frames attached to the robot links and using homogeneous transformation matrices. (used for the project, define frames on each links of the robot, you need to fine the different homogeneous transformation matrices and multiply them to obtain the position and the orientation of the end effector given the joint variable).

When we are dealing with more than 2 or 3 joints not in a planar case we need a systematic method in order to define which is the position and orientation of the end effector with respect to the base B frame: T .E

We need to identify a frame for each joint in the manipulator, and then we will find all the different 1st, 2nd, 3rd T matrices from the 1st frame to the 2nd frame, from the 2nd to the 3rd and so on. Then, we can multiply all of them an obtain the final matrix from the end effector to the base. One way to find these frame and t matrices is DENAVIT-HARTENBERG CONVENTIONS (SYSTEMATIC METHOD).

The convention defines only 4 parameters between two successive RF,

instead of the usual 6,because two constraints are added. This T matrix with this method is completely defines by the useof 4 parameters: 2 parameters are associated with a translation, 2 parameters are associated witha rotation. Three of these parameters depend on the robot geometry only, and therefore areconstant in time. One parameter depends on the relative motion between two successive links,and therefore is a function of time. Parameters are called the i-th joint variable q (t)i

13DIRECT AND INVRESE KINEMATICS OF THE MANIPULATORDENAVIT-HARTENBERG (DH) FRAMES – find the four parameter over consecutive frames to findthe T matrix.

1. Define all the joint axis on the robot, in particular in case of revolute joint, identify all the axis of rotation.
2. Once find the rotation axis, compute the common normal to actual joint axes and previous one.Remember that the common normal the a line perpendicular to both rotational axes.
3. The reference frame is putted over the next joint axis

In order to consider the complete movement of the joint because the rotation of joint 1 affect the rotation of FR1 (putted in the second joint).

The z-axis is perfectly aligned with the axis of rotation.

The x-axis instead for the first and last frame is completely arbitrary, while for all the other frame is aligned with the common normal.

Following the right handed rule find the y-axis.

Also the centre of the first and last frame (basis frame and end effector frame) is arbitrary.

SPATIAL RELATION BETWEEN JOINT AXES (2 parameters)

1. a (or r): displacement AB between joint axes (i.e. the length of the common normal).
2. α: twist angle between joint axes (i.e. the angle around the common normal to between the previous rotation axis and current rotation axis)

When there are 2 parallel consecutive joint axes (2D exercise) the length a is exactly the length of the link, and α is equal to zero (we have no twist).

SPATIAL RELATION BETWEEN LINK AXES

d = displacement DC (a

variable θ = angle between link axes (a variable if joint is revolute): in case of prismatic joint the angle around the z-axis between the previous x-axis and the current x-axis.

When there are 2 parallel consecutive joint axes (2D exercise) θ is equal to q (revolute joint) while 2d, since we have no displacement between the axis, is 0.

DENAVIT-HARTENBERG PARAMETERS

- unit vector z along axis of joint i+1

- unit vector x along the common normal to joint I and i+1 axes (i → i+1)

- a = distance DO – positive if oriented as x (constant = 'length')

of link i):

• i: index of the joint
• d: distance O D - positive if oriented as z (variable if joint is prismatic)

i-1:

• i-1: index of the previous joint
• α: twist angle between z and z around x (constant)

i:

• i: index of the current joint
• θ: angle between x and x around z (variable if joint is revolute)

AMBIGUITIES IN DEFINING DH FRAMES

Frame (first frame): origin and x axis are arbitrary

Frame (the end effector): z axis is not specified (x must be orthogonal to and intersect z)

n:

• n: index of the next joint
• n-1: index of the previous joint

Positive direction of z (up/down on joint i) is arbitrary (z axis is the axis around with the frame i-1 rotates or along which the frame translates). Choose one, and try to avoid "flipping over" to the n