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Industrial Robots Programming - J. Norberto Pires Part 4

Tham khảo tài liệu 'industrial robots programming - j. norberto pires part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Robot Manipulators and Control Systems 51 Substituting in 2.37 bD bR 1 AD results in aVd ỒR BR AD 2.38 Because jR is an orthonormal matrix we can write 1 7 BR A 1 bS 2.39 where bS is a skew-symmetric matrix associated with bR . Using 2.39 in 2.38 gives aVd aD 2.40 The skew-symmetric matrix bS defined in 2.39 is called angular velocity matrix. Writing s as 2.41 and the vector Q 3x 1 as n A A A. results in 2.42 -íìzDy ĩì y D z ÍĨZDX -QXDZ OyDx QxDy Í2 X D 2.43 where D DX Dy Dz T is a position vector. The vector ÍÌ associated with the angular velocity matrix is called an angular velocity vector. Using 2.43 and 2.40 gives aVd aQb X AD 2.44 52 Industrial Robots Programming Considering now the linear and angular accelerations of each link it s possible to write by direct differentiation of 2.34 AVD AVB R bVd aQbx Ar nD AQBx AR bD 2.45 or since BR bVd BRB D Aí2b X bRBVd and bR bD gR bVd aQb X bR BD aVd aVb r BVD 2AQnx Ar bVd A Q B X Ar bD aQb X aQb X Ar bD 2.46 The above equation is the general equation for the linear acceleration of point D about A and expressed in terms of A . If BD is a constant vector like in robotics applications then equation 2.46 simplifies to AVD AVB A0Bx r bD aQb X aQb X Ar bD 2.47 because BVD B V D 0. If we consider a third frame C with AQfi being the angular velocity of B about A and Bíỉc the angular velocity of B about C then the angular velocity of C about A is aQc aQq Ar bQc 2.48 Taking the derivative of 2.48 results in Anc A 0b r Bnc Ai2B bR Bíìc aí2bx bR bQc 2.49 This is a very useful equation to compute the angular acceleration propagation from link to link. Let s apply this to a robot manipulator. As mentioned before we will consider only rigid manipulators with revolutionary joints with the base frame as the reference frame. Robot Manipulators and Control Systems 53 zi l f I Axis i Figure 2.10 Linear and angular velocity vectors of adjacent links The angular velocity of link i 1 expressed in terms of i is given by5 wi I wj i l R éi ii .