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Mobile Robots Perception & Navigation Part 2

Tham khảo tài liệu 'mobile robots perception & navigation part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Visually Guided Robotics Using Conformal Geometric Computing 31 surface is a surface generated by the displacement of a straight line called generatrix along a directing curve or curves called directrices . The plane is the simplest ruled surface but now we are interested in nonlinear surfaces generated as ruled surfaces. For example a circular cone is a surface generated by a straight line through a fixed point and a point in a circle. It is well known that the intersection of a plane with the cone can generate the conics. See Figure 10. In 17 the cycloidal curves can be generated by two coupled twists. In this section we are going to see how these and other curves and surfaces can be obtained using only multivectors of CGA. Fig. 10. a Hyperbola as the meet of a cone and a plane. b The helicoid is generated by the rotation and translation of a line segment. In CGA the motor is the desired multivector. 5.1 Cone and Conics A circular cone is described by a fixed point v0 vertex a dual circle z0 a0 A a1 A a2 directrix and a rotor R 0 l 0 e 0 2 rotating the straight line L v0 a0 v0 A a0 A e v generatrix along the axis of the cone l0 z0 ex. Then the cone w is generated as I.-.6.-J rn.-ũ I 67 A conic curve can be obtained with the meet 17 of a cone and a plane. See Figure 10 a . 5.2 Cycloidal Curves The family of the cycloidal curves can be generated by the rotation and translation of one or two circles. For example the cycloidal family of curves generated by two circles of radius r0and r1 are expressed by see Figure 11 the motor M TRiTR 68 where T T r0 n sin ỡ ei cos 0 e2 69 70 R2 R2 ỡ 71 32 Mobile Robots Perception Navigation Then each conformal point x is transformed as MxM. 5.3 Helicoid We can obtain the ruled surface called helicoid rotating a ray segment in a similar way as the spiral of Archimedes. So if the axis e3 is the directrix of the rays and it is orthogonal to them then the translator that we need to apply is a multiple of 0 the angle of rotation. See .