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3 KINEMATICS – SERIAL ROBOT In the book Introduction to Robotics Mechanics & <strong>Control</strong>, kinematics is described as follows: “Kinematics is the science of motion which treats motion without regard to the forces which cause it. Within the science of kinematics one studies the position, velocity, acceleration and all higher order derivatives of the position variables, with respect to time or any other variable(s)” (2). The kinematics for a parallel manipulator is too complicated to be present at this point. Therefore we will start with a two-link robot in two-dimension as in Spong and Vidyasagar (5) to give the reader more background how the kinematics works for an industrial robot. The kinematic for a Delta-3 Robot in three dimensional is calculated in chapter 4. The kinematics for an industrial robot can be distributed in three different problem formulations, Forward Kinematics, Inverse Kinematics and the Velocity Kinematics. Consider Figure 3.1 as a starting-point for the introduced kinematics for a two-link serial industrial robot. 3.1 FORWARD KINEMATICS Figure 3.1, two-link serial robot with end-effector at starting point. The first problem is to describe both the position of the tool and the start and end location (also the surface between these two locations) with respect to a base frame, reference coordinate system. Most commonly is to establish a fixed coordinate system, base frame, to which all objects including the manipulator are referenced. Then each new coordinate system for each joint can be expressed in the base frame with help by transformation matrixes (rotation matrixes). ______________________________________________________________________________ Public Report ELAU GmbH, Marktheidenfeld 10
Usually, the manipulator will be able to sense its own joint angles using some type of resolvers or encoders for each angle. With help of these angles the end-effector or tool position can be determined by the Forward Kinematics, which is to determine the end-effector or tool position in terms of the joint variables. In this case of a two-linked robot manipulator the base coordinate frame o0x0y0 is established at the base of the robot as showed in Figure 3.2. Then the coordinates (x, y) of the tool can be expressed in the base frame with the two equations shown in Eq. 3.1 below. ⎧x = a cos( θ1) + bcos( θ2 ) ⎨ ⎩y = asin( θ1) + bsin( θ2 ) Eq. 3.1 Where a is the length of the lower arm, b is the length of the upper arm and θ1,θ2 are the angles for lower and upper arm respectively as shown in Figure 3.2. Figure 3.2, two-link serial robot. Figure 3.3, frame transformation between lower arm and tool frame. The orientation of the joint frame between the lower and the upper arm, and the tool frame can be described with help of a transformation matrix relative the orientation of the base frame. Consider Figure 3.3 where i0 and j0 are orthonormal unit vectors in the base frame, and i2 and j2 are orthonormal unit vectors in the tool frame. Then the rotation of the tool frame can be described by an orientation matrix as in Eq. 3.2. r r ⎡ i0 ⋅ i2 ⎢r r ⎣i0 ⋅ j2 r r j ⋅ ⎤ ⎡cos( + ) 0 i θ1 θ 2 2 r r ⎥ = ⎢ j ⋅ ⎦ ⎣sin( θ1 + θ 2 ) 0 j0 Eq. 3.2 − sin( θ1 + θ 2 ) ⎤ cos( θ + ) ⎥ 1 θ 2 ⎦ These equations Eq. 3.1- Eq. 3.2 are called the forward kinematics equations in (5 p. 21) . But for a 3-DOF parallel manipulator these equations can be calculated so that only one angle is needed for each arm structure instead of two angles as for the two linked robot. This makes the equations a bit more complex and cannot be written as easily as for a two-linked manipulator. ______________________________________________________________________________ Public Report ELAU GmbH, Marktheidenfeld 11
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 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 26 and 27: where & ⎡x& n ⎤ ⎡− lA sin(
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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