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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion A manipulator can be modelled as a kinematic chain made of rigid bodies (links) connected by joints (rotoidal or prismatic) For each joint correspond a degree of freedom Cinematica dei Manipolatori e dei robot mobili In manipulators one extreme is rigidly constrained to a base, to the other end is fixed an end effector. Knowing the variable associated with each joint (joint variable: angle for revolute, relative position for prismatic), the position and orientation (pose or posture) of the terminal are uniquely determined In the case of mobile robots we do not have rigid constraints to determine the position unequivocally, but wheels (generally) that determine a speed constraint Knowing the variable connected with each joint (total angle of rotation of the wheel) there is no information about the pose of the vehicle

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Cinematica dei Manipolatori In manipulators the overall motion is realized by means of the composition of the relative motions of each arm with respect to the previous In the operations of manipulation is necessary to describe the position and orientation of the end effector (pose), in the case where there are obstacles it is necessary to consider position and orientation of each joint (posture) Regarding the kinematics of manipulators we will consider: - the direct kinematics by means of which, as a function of the variables of the joint, is possible to derive the end effector pose with respect to the reference system - the working space and the joint space for the management of the workspace of the robot - the calibration of kinematic parameters in order to obtain a better positioning accuracy - the solution of inverse kinematics by which we determine the joint variables from the end effector pose

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Cinematica dei Manipolatori A rigid body is completely described by means of the pose, or by the position of a reference point and the attitude of the body reference system wrt the reference Dove: o x o y o z vector o along the reference system x y z unit vector wrt the reference system o = o x x + o y y + o z z

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Cinematica dei Manipolatori The vector o is an applied vector since, in addition to the direction and module, it is also specified the application point O- x y z reference O- x y z body reference Wrt reference system the unit vectors of the body reference are expressed by the equations:

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Cinematica dei Manipolatori O- x y z absolute reference O- x y z body reference The orthonormality allows the above relation that also implies: … that is a simple scalar product The body reference unit vectors decomposition wrt the absolute reference:

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Cinematica dei Manipolatori The three vectors can be combined to form a matrix: And therefore: R is defined ortogonal, and holds: The vectors of the columns of R build an orthonormal reference so that: R provides the orientation of the body reference wrt the absolute reference

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Cinematica dei Manipolatori – Trasformazione coordinate If the body reference origin coincides with the absolute reference origin: o = 0 (zero) P can be represented wrt both references:

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion P and p are representations of the same vector: Note: x, y, z are the unit vectors of wrt, i.e. we observe from the absolute reference So R is the coordinates transformation matrix of the vector expressed wrt in the coordinates of the same vector expressed with respect to and vice versa via R T Cinematica dei Manipolatori – Trasformazione coordinate

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Cinematica dei Manipolatori – Rotazione di un vettore p is always identical whichever motion (are the body reference coordinates), p varies as a function of the rotation providing the vector wrt the absolute reference So R represent the rotation matrix of the vector p from the position solidal to to the actual one If p' is the representation of the vector with respect to the body reference, if we assume that the initial conditions of the reference and the solidarity reference coincide: p' = p at the beginning. Following a pure rotation (still suppose o'= 0) of the body:

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion At the beginning the two references coincide and so p = p After the rotation p = R p p p Nota: rotation preserve modulus: Cinematica dei Manipolatori – Rotazione di un vettore p = p p = R p

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion R resembles three different geometrical meanings: 1.the orientation of one reference wrt another 2.Coordinates transformation 3.Rotation matrix Cinematica dei Manipolatori – significato di R

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion C. M. – composizione matrici di trasformazione coordinate Given 3 references: O-x 0 y 0 z 0, O-x 1 y 1 z 1, O-x 2 y 2 z 2, with the same origin A vector p can be expressed wrt each of the three p 0 p 1 and p 2 If we define R i j the matrix of coordinates transformation from i to j ( j is the reference, the one wrt the coordinates are expressed) The matrix that expresses the coordinates of a point with respect to the reference 0 starting from those expressed with respect to 2 is equal to the composition of two partial transformations: transformation from 2 to 1 transformation from 1 to 0 The overall transformation matrix is determined by multiplying from right to left

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion C. M. – composizione matrici di rotazione (terna corrente) Starting with a reference superimposed to the 0 The rotation expressed by R 2 0 (rotation from 0 till 2: note that from the point of view of the rotation sense is from the apex to the subscript) we obtain: 1.Rotation from 0 to 1 (current reference 0) 2.Rotation from 1 to 2 (current reference 1) The composition of rotations from 0 to 2 are obtained multiplying from left to right Since each partial rotation is obtained from the outcome of the previous one, the rotation reference is called the current reference The operation is defined "successive rotations wrt current reference axes

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Note: the result depends from the order of rotation C. M. – composizione matrici di rotazione (terna corrente)

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Rotazioni rispetto alla terna corrente - Esempio

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Rotazioni rispetto alla terna corrente - Esempio Comp sx toward dx

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion C. M. – operazioni di rotazione rispetto a terna fissa In this case the multiplication is from right to left If the rotations are defined wrt the absolute (fixed) reference then the operation is defined "successive rotations wrt fixed reference axes Suppose we have a point p that rotate: 1. first according to a rotation matrix defined by R 1 2. then according to a rotation matrix defined by R 2 The resulting vector p'' is given by:

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion C. M. – operazioni di rotazione rispetto a terna fissa

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Rotazioni rispetto alla terna fissa - Esempio Comp dx verso sx

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion C. M. – Trasformazioni Omogenee In this case the homogeneous representation is used: If we have also a translation So we define a,matrix of homogeneous transformation: So the coordinates transformation(from reference 1 to 0) can be written:

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion C. M. – Trasformazioni Omogenee From 0 to 1 : where: Note :

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Verification:

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion As for the coordinates transformation we have a product from right to left : C. M. – Trasformazioni Omogenee Comp dx verso sx

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