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where & ⎡x& n ⎤ ⎡− lA sin( θi ) ⎤ R = ⎢ ⎥ i R ⎢ 0 ⎥ ⎢ y& − & n ⎥ ⎢ ⎥ θ = X& − b & i n iθi i = 1, 2, 3 ⎢⎣ z& ⎥⎦ ⎢⎣ l cos( ) ⎥ n A θi ⎦ si z For one robot arm the Eq. 4.7 can be written as Eq. 4.8 ⎡− lA sin( θi ) ⎤ R b R ⎢ i z 0 ⎥ i = ⎢ ⎥ i = 1, 2, 3 ⎢⎣ l cos( ) ⎥ A θi ⎦ s T i Eq. 4.9 ⎡xn ⎤ ⎡0⎤ ⎢ ⎥ T s ⎢ i 0 ⎥ ⎢ yn ⎥ − biθi = ⎢ ⎥ ⎢⎣ z ⎥⎦ ⎢⎣ 0⎥ n ⎦ & & & & Eq. 4.10 i = 1, 2, 3 which can be expressed in matrix form for all of the three robot arms as T T ⎡s ⎤ ⎡ 1 s1 1 ⎢ T⎥ ⎢ ⎢s2 ⎥ − ⎢ 0 ⎢ T⎥ ⎢ ⎣s3 ⎦ ⎣ 0 & b X n s 0 ⎤ ⎡0⎤ ⎥ 0 ⎢ ⎥ ⎥θ = ⎢ 0 ⎥ T s ⎥ ⎦ ⎢⎣ 0⎥ 3 3 ⎦ & b ______________________________________________________________________________ Public Report ELAU GmbH, Marktheidenfeld 20 0 T 2 b 0 2 Eq. 4.11 From Eq. 4.11 the Jacobian matrix for a Delta-3 robot is obtained by considering where −1 θ& & X J n = Eq. 4.12 T T ⎡s ⎤ ⎡ 1 s1 b ⎢ T ⎥ ⎢ J = ⎢s 2 ⎥ ⎢ 0 ⎢ T ⎥ ⎢ ⎣ s3 ⎦ ⎣ 0 1 Eq. 4.13 s 0 T 2 b 0 2 0 ⎤ ⎥ 0 ⎥ T s ⎥ 3b3 ⎦ The Jacobian matrix J is not only depending of θ, as is usually the case for serial robots, but also a function of the TCP position Xn, which can be calculated with the forward kinematic model of the Delta-3 robot.
4.6 ACCELERATION KINEMATICS The acceleration kinematics specifies a mapping from acceleration in joint space to acceleration in Cartesian space. To calculate this mapping one can use the result from the previous section 4.5 and derive Eq. 4.11 one more time to obtain the acceleration relationship. It is worth noting here that the time derivative of Eq. 4.11 is the derivative of a product. Deriving Eq. 4.11 gives T T ⎛ T ⎡s ⎤ ⎡s& ⎤ ⎡ 1 1 ⎥ ⎜ s1 b1 ⎢ T ⎥ ⎢ T ⎢ ⎢s ⎥ + ⎢ ⎥ − ⎜ 2 X& & s& & n 2 X n ⎢ 0 ⎢ T ⎥ ⎢ T ⎥ ⎜ ⎢ ⎣ s3 ⎦ ⎣ s& 3 ⎦ ⎝⎣ 0 s 0 T 2 b 0 2 T T 0 ⎤ ⎡s& 1 b1 + s1 b& ⎥ ⎢ 0 & ⎥θ& + ⎢ 0 T s ⎥ ⎢ 3b3 ⎦ ⎣ 0 Eq. 4.14 ______________________________________________________________________________ Public Report ELAU GmbH, Marktheidenfeld 21 1 0 s& b + s b& T 2 2 0 T 2 2 0 ⎤ ⎞ ⎟ ⎡0⎤ ⎥ 0 &⎟ = ⎢ ⎥ ⎥θ ⎢ 0 ⎥ T T ⎥ ⎟ s& + s ⎦ ⎢⎣ 0⎥ 3b3 3b& 3 ⎠ ⎦ Using the definition of Eq. 4.12 and Eq. 4.13 and rearranging Eq. 4.14, the following is obtained T −1 ⎛ T T T ⎡s ⎤ ⎡& ⎤ ⎡& b b& ⎤⎞ 1 ⎜ s1 s1 1 + s1 1 0 0 ⎟ X& & ⎢ T⎥ ⎢ T⎥ ⎢ T T ⎜ & J & b b& ⎥ ⎟ & θ J & θ& n = ⎢s2 ⎥ − ⎢s2 ⎥ + ⎢ 0 s2 2 + s2 2 0 ⎥ + ⎢ T⎥ ⎜ ⎢ T & ⎥ ⎢ T T & b b& ⎥ ⎟ ⎣ s3 ⎦ ⎝ ⎣ s3 ⎦ ⎣ 0 0 s3 3 + s3 3⎦ ⎠ Eq. 4.15 In Eq. 4.15, the time derivative of the Jacobian matrix J& can be identified as the term multiplying θ& . And finally, the relationship between the Cartesian acceleration and the acceleration in joint space can be expressed as X & = J& & θ + J & θ& n Eq. 4.16
Page 1: ISSN 0280-5316 ISRN LUTFD2/TFRT--58
Page 5: Acknowledgements This Master thesis
Page 8 and 9: 7 Control approach - ELAU GmbH ....
Page 10 and 11: α1 frame rotation angle for arm at
Page 12 and 13: 2 BACKGROUND 2.1 ROBOTICS The word
Page 14 and 15: 2.2.2 SOFTWARE - ELAU GMBH EPAS-4 i
Page 16 and 17: 3 KINEMATICS - SERIAL ROBOT In the
Page 18 and 19: 3.2 INVERSE KINEMATICS When the for
Page 20 and 21: 4 KINEMATICS & KINETICS - DELTA-3 R
Page 22 and 23: 4.3 Forward Kinematics The forward
Page 24 and 25: Figure 4.5, inverse kinematics tran
Page 28 and 29: 4.7 ACTUATOR DYNAMICS In most exist
Page 30 and 31: The inertia contribution of each up
Page 32 and 33: According to d’Alembert’s princ
Page 34 and 35: 5 TRAJECTORY A trajectory is a path
Page 36 and 37: 6 TRAJECTORY - ELAU GMBH C600 is th
Page 38 and 39: 6.2 CIRCULAR INTERPOLATION In the E
Page 40 and 41: 7.1 MC-4 DRIVE The MC-4 drive is a
Page 42 and 43: Figure 8.2, Simulink model with mot
Page 44 and 45: 9 PARAMETER ESTIMATION To simulate
Page 46 and 47: 10 EXPERIMENTS For the experiments
Page 48 and 49: Figure 11.2, shows the motion of th
Page 50 and 51: Figure 11.6, shows the current I ef
Page 52 and 53: 11.1.2 SIMULINK EXPERIMENT 2 Figure
Page 54 and 55: Figure 11.13, shows the acceleratio
Page 56 and 57: measured actual value from the real
Page 58 and 59: and are shown over the time in seco
Page 60 and 61: Figure 11.23 shows the x, y and z c
Page 62 and 63: Figure 11.27, the blue line shows t
Page 64 and 65: Figure 11.31, the blue line shows t
Page 66 and 67: In Figure 11.36 the blue line repre
Page 68 and 69: 11.2.3 JACOBIAN EXPERIMENT 3 Figure
Page 70 and 71: In Figure 11.44 the blue line repre
Page 72 and 73: 12 CONCLUSIONS In section 11.1 one

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