A QUICK SKETCH OF MAJOR ISSUES IN ROBOTICS in Visual Studio .NET

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A QUICK SKETCH OF MAJOR ISSUES IN ROBOTICS
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link i center of mass r* i origin of link i + 1, joint i
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origin of link i, joint i 1
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p* i
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ri pi 1
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arm base
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Balance of forces and torques acting on a single link.
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the accelerations r1 and r2 of the links centers of mass by Newton s second law, f1 = m1 r 1 f2 = m2 r 2 (2.9)
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From these equations, accelerations ri of the centers of mass can be derived. Let i be the angular velocity vector of the center of mass of link i. Let i be the corresponding angular acceleration. Let Ii be the inertia matrix of link i. Then torques are related to angular velocities and accelerations by Euler s equations, n1 = I1 1 + 1 I1 1 n2 = I2 2 + 2 I2 2 (2.10)
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For our planar two-link manipulator shown in Figure 2.1, the torque is normal to the arm s plane. Rotary inertia through the centers of mass of links 1 and 2 are [7]
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2 I1 = m1 l1 /12 + m1 R 2 /4 2 I2 = m2 l2 /12 + m2 R 2 /4
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(2.11)
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DYNAMICS
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Angular velocities and accelerations are 1 = 1 2 = 1 + 2 1 = 1 2 = 1 + 2
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(2.12)
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Substituting those into Euler s equations (and taking into the account that i Ii i = 0), we obtain n1 = I 1 1 n2 = I2 ( 1 + 2 ) (2.13)
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Finally, Newton Euler equations are combined with static equations [Eq. (2.8)] to produce the torques at arm joints that is, to do inverse dynamics. After simpli cations, these become (details can be found in Refs. 7 and 8) n1,2 = 1 I2 + +
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2 m2 l2 m2 l1 l2 cos 2 + 2 4
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+ 2
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2 m2 l2 4
m2 l1 l2 2 m2 l2 g2 1 sin 2 + cos ( 1 + 2 ) 2 2 (2.14)
l2 sin ( 1 + 2 )f2,3x l2 cos ( 1 + 2 )f2,3y + n2,3 n0,1 = 1 I1 + I2 + m2 l1 l2 cos 2 + + 2 I2 + +
2 2 m1 l1 + m2 l2 2 + m2 l1 4
2 m2 l2 m2 l1 l2 + cos 2 4 2
m2 l1 l2 2 2 sin 2 m2 l1 l2 1 2 sin 2 2 m1 m2 l2 cos ( 1 + 2 ) + l1 + m2 2 2 cos 1 g2
(l1 sin 1 + l2 sin( 1 + 2 ))f2,3y + n2,3 There are three types of terms that appear in such equations. Taking as an example the above equation for n1,2 , these are: Dynamic Torques (Terms 1, 2, and 3). These arise from the arm movement, and depend on velocities and accelerations. Gravity Torques (Term 4). These are due to the (vertical) gravity force.
A QUICK SKETCH OF MAJOR ISSUES IN ROBOTICS
External Torques (Terms 5, 6, and 7). These are due to external forces and torques that come from the arm s interaction with other objects; they appear when the arm physically touches objects, such as in assembly or cleaning. During a free arm motion these torques are zeros. Then there is another classi cation of dynamic torques, also with three types: Inertial Torques. These are proportional to accelerations in the arm joints, and they arise from normal action/reaction forces of an accelerating body. Centripetal Torques. These torques arise from a constrained rotation about a point, and they are proportional to the squares of joints velocities. For example, the arm s forearm must rotate about the arm s shoulder joint, and so the centripetal acceleration is aimed at the shoulder joint along link l1 (Figure 2.1). Coriolis Torques. These torques arise from vortical forces, as a result of interaction between two rotating systems (in our case two arm links), and they are proportional to the product of joint velocities of two different links. Notice the remarkable growth in the complexity of equations as we proceed from kinematics to statics to dynamics equations. Then there is another natural source of complexity the arm complexity, measured by the number of robot degrees of freedom. The reader is reminded that in our example, Figure 2.1, we are dealing with the simplest two-link planar manipulator: In its analysis we started with modest kinematic equations (2.2) and (2.3) and arrived at rather complex dynamic equations in (2.14). Will the equations complexity grow as quickly with the growth in the number of robot DOF Indeed they will. As an example, if we write only the acceleration-related coef cients for an arm with six DOF, they form this 6 6 matrix (note that a great many of today s industrial robot arm manipulators have six or more DOF): D11 D12 D13 D14 D15 D16 D12 D22 D23 D24 D25 D26 D D D D D D D = 13 23 33 34 35 36 (2.15) D14 D24 D34 D44 D45 D46 D D D D D D 15 25 35 45 55 56 D16 D26 D36 D46 D56 D66 The diagonal terms in this matrix represent uncoupled terms that is, terms caused by a single joint and off-diagonal terms represent pairwise interaction effects for all six joints. Each such term is itself a rather complex relationship. Out of curiosity, if the very rst term in the matrix above, D11 , is written in full, it looks as shown in Figure 2.6 [9]. As you glance at this formula, try to imagine what the whole matrix D must look like, and imagine what kind of complexity a control system based on such expressions must involve. Consequently, all kind of simpli cations are done in real-world systems when designing robot control schemes. Simpli cations in equations bring, of course, imprecision, and so the design process involves carefully studied trade-offs.