Proceedings of the World Congress on Engineering 7 Vol II WCE 7, July - 4, 7, London, U.K. Robot Motion Planning using Hyerboloid Potential Functions A. Badawy and C.R. McInnes, Member, IAEN Abstract A new aroach to robot ath lanning using hyerboloid otential functions is resented in this aer. Unlike arabolic otential functions, where the control force increases with distance from the goal and is unbound, and conic otential functions where a singularity occurs at the goal, hyerboloid otential functions aoid both these drawbacks. Howeer, they do combine the adantages of both arabolic and conic otentials as the asymtotic roerty of the hyerbolic function ensures bounded control forces, while stability and smooth contact are guaranteed at the goal oint. Index Terms Obstacle aoidance, otential functions, robot motion lanning. I. INTRODUCTION Potential field methods roide an elegant, simle and comutationally efficient aroach for single and multile robot motion lanning. The method was introduced first as a motion lanning algorithm for maniulator arms [], and then generalized for robot motion lanning in dynamic enironments []. Sace alications of the otential field method hae been deeloed for arious tasks such as formation-flying [3], roximity manoeuring [4], and on-orbit assembly [5], [6]. The otential field method constructs an attractie otential field that is resonsible for directing a manoeuring object toward its goal configuration. Collisions with other objects or obstacles in the worksace are aoided through constructing high otential fields surrounding them. Various functional forms of reulsie otential fields hae been inestigated, including: FIRAS [], suerquadric functions [7], naigation functions [8], aussian distributions and ower laws [9], and the Lalace equation []. This aer introduces a new reresentation for the attractie otential field using hyerbolic functions. They roide key adantages oer araboloid or conic functions. In addition, suerquadric obstacle otentials will be used where the orientation of the suerquadric is defined using quaternions. As will be seen, rotational and translation motion then becomes strongly couled, allow efficient manoeuring []. Manuscrit receied February 7, 7. A. Badawy is with the Uniersity of Strathclyde, lasgow, UK, (hone: 4 548 485; fax: 4 55 55; e-mail: ahmed.badawy@strath.ac.uk. Colin R. McInnes is with the Uniersity of Strathclyde, lasgow, UK (e-mail: colin.mcinnes@strath.ac.uk. II. TRANSLATIONAL ATTRACTIVE POTENTIAL A manoeuring object is stimulated to moe toward its goal configuration through an attractie otential field. Any function could be utilized roiding it satisfies Lyauno's stability conditions and its global minimum is laced at the goal configuration. Preiously, two main tyes of attractie otential hae been used: arabolic and conical []. The new hyerbolic function roides bounded control action while also roiding smooth motion in the neighbourhood of the goal configuration. A. Parabolic Attractie Potential The arabolic function is commonly used in motion lanning roblems, as shown in Fig.. A manoeuring object is defined with the osition, r, and elocity, r&, and is required to moe to a goal oint at osition, r, with goal elocity, r&. Defining the otential function as [3]: Vatt,trans + r& where and are constant gain factors, the time deriatie of the arabolic otential function is exressed as: ( ( r & ( + (&& r att,trans. ( ( To ensure stability using Lyauno's second theorem the time deriatie should be non-ositie eerywhere in the worksace since the otential function is ositie definite. A suitable control law is then exressed as: & ( for some control gain. As the distance between the manoeuring object and its goal increases, the required control force to ensure stability increases and is unbound. Hence, for real systems actuator saturation may occur and the stability of the roblem is not guaranteed. To circument this roblem a conical otential function can be used. (3 ISBN:978-988-9867--6 WCE 7
Proceedings of the World Congress on Engineering 7 Vol II WCE 7, July - 4, 7, London, U.K. C. Hyerbolic Attractie Potential Continuous control of a mobile robot is carried out either by merging the arabolic and conical otentials each oer a certain range, or now by using a hyerbolic function, as shown in Fig. 3. Near its global minimum, the hyerbolic function has a smooth shae like the arabola, while away from the minimum it becomes asymtotic with constant gradient like the conical field. The hyerbolic otential is described as: Vatt,trans + r& (7 The time deriatie is then exressed as: Fig. Parabolic-well attractie otential B. Conical Attractie Potential The conical attractie otential does not hae the difficulties discussed aboe, as shown in Fig.. It can be exressed as [3]: Vatt,trans + r& (4 The time deriatie of the otential is then exressed as: (5 ( r & r att,trans. Therefore, a suitable control law is then gien by: & r & The control force remains bound regardless of the location of the manoeuring object. Howeer, the goal oint is singular. (6 att,trans. ( r & r Finally, a suitable bounded, smooth and singularity-free control law is exressed as: & The control law defined in Eq. (9 is significantly better than those defined in Eqs. (3 and (6 as the control force remains bounded whateer the distance to goal is, and the singularity at the goal is remoed. Hence, global conergence is achieed. The single hyerbolic function is also more comutationally efficient to imlement comared to the use of a combined conical and arabolic otential with a switching function in the neighbourhood of the goal. Again, the single function contains the key features of both the arabolic and conical fields. (8 (9 Fig. Conic-well attractie otential Fig. 3 Hyerbolic-well attractie otential ISBN:978-988-9867--6 WCE 7
Proceedings of the World Congress on Engineering 7 Vol II WCE 7, July - 4, 7, London, U.K. III. LOBAL POTENTIAL FUNCTION Translational motion of the manoeuring object to the goal is not the sole objectie of the otential function. Orientation is also of imortance for extended rigid bodies manoeuring to some goal orientation. In addition, collision aoidance with other manoeuring objects and obstacles is a key requirement of the global otential function which is exressed as: V + + qq. q + ω ω. ω + Vobs r& where q and ω are constant gains, q q q T q 3 ( is the quaternion ector reresenting orientation, omitting the forth term q 4, ω is the angular elocity ector, and V obs is the obstacle otential field. The time deriatie of the global hyerbolic otential is then exressed as: + Vobs + ( q qq&. q + ω&. ω + q& ω. Vobs. r & and & Vobs (5-a q ω& q4q + ωω + Q q Vobs (5-b ω ω The control laws defined by Eq. (5 are of a general form for any reulsie otential. Suerquadric obstacle otentials are chosen for the remainder of the aer []. These functions roide a means of caturing the geometric shae of extended rigid bodies and allow couled translational and rotational motion lanning. IV. NUMERICAL RESULTS The hyerbolic otential field is used with continuous control to assemble seen beam elements to form a truss structure. Objects are initially laced arallel to the z-axis, Fig. 4. The assembly of the objects is demonstrated in Fig. 5, where Fig. 6 shows the eolution of the object dynamics. Noting the log-scale, it can be seen that a smooth acceleration- coastbraking rofile is generated by the control law. with x y z T and q q q T q 3. Equation ( can be simlified using the following relations: q& Qω ( z while q4 q3 q Q q3 q4 q (3 q q q4 Fig. 4 Initial object configuration y x Finally, the time deriatie can be exressed as q ω. qqq + ω ω& + Q Vobs. r & + Vobs + (4 Hence, the control laws are exressed as: Fig. 5-a Object configuration (t 37 sec ISBN:978-988-9867--6 WCE 7
Proceedings of the World Congress on Engineering 7 Vol II WCE 7, July - 4, 7, London, U.K..5.4.3 3 4 5 6 7. Velocity in x-direction [m/sec]. -. -. -.3 -.4 -.5-3 - - Fig. 5-b Object configuration (t sec Fig. 6-a Object elocities in x-direction.3.5. 3 4 5 6 7 Velocity in z-direction [m/sec].5..5 -.5 - - Fig. 5-c Object configuration (t 8 sec Fig. 6-b Object elocities in z-direction. Angular elocity about y-axis [rad/sec].8.6.4. -. -.4, 5, 4, 6 3, 7 -.6 - - Fig. 5-d Final object configuration (t 3 sec Fig. 6-c Object angular elocities about y-axis ISBN:978-988-9867--6 WCE 7
Proceedings of the World Congress on Engineering 7 Vol II WCE 7, July - 4, 7, London, U.K. V. CONCLUSION Adding a elocity term to a hyerbolic otential function roides successful continuous control with bounded control action. The resulting controlled elocities are nearly constant oer the entire worksace, excet in the neighbourhood of obstacles. lobal stability and conergence of the system is roen and tested for a dense worksace. Proximity motion of the manoeuring objects shows the couling between translational and rotational motion in the resence of obstacles. REFERENCES [] O. Khatib, "Real-time obstacle aoidance for maniulators and mobile Robots", The International Journal of Robotics Research, Vol. 5, No., 986,. 9-98. [] S.S. e, and Y.I. Cui, "Dynamic motion lanning for mobile robots using otential field method, Autonomous Robots, Vol. 3,,. 7-. [3] F. McQuade and C.R. McInnes, Autonomous control for on-orbit assembly using otential function methods, The Aeronautical Journal, Vol., No. 8, 997,. 55-6. [4] C.R. McInnes, "Autonomous roximity maneuering using artificial otential functions", ESA Journal, Vol. 7, 993,.59-69. [5] D. Izzo, L. Pettazzi, and M. Ayre, "mission concet for autonomous on orbit assembly of a large reflector in sace", IAC-5-D.4.3, 56th International Astronautical Congress, Fukoka, Jaan, Setember, 5. [6] A. Badawy, and C.R. McInnes, "Autonomous structure assembly using otential field functions", IAC-6-C.P.3.4, 57 th International Astronautical Congress, Valencia, Sain, October, 6. [7] R. Vole and P. Khosla, "Maniulator control with suerquadric artificial otential functions: Theory and Exeriments", IEEE Transaction on Systems, Man, and Cybernetics, Vol., No. 6, 99. 43-436. [8] E. Rimon and D.E. Kodischek, "Exact robot naigation using artificial otential functions", IEEE Transactions on Robotics and Automation, Vol. 8, no. 5, 99,. 5-58. [9] F. McQuade, "Autonomous control for on-orbit assembly using artificial otential functions", Ph. D Dissertation, Uniersity of lasgow, 997. [] A.B. Roger and C.R. McInnes., Passie-safety constrained free-flyer ath-lanning with Lalace otential guidance at the International Sace Station, Journal of uidance, Control and Dynamics, Vol. 3, No. 6,,. 97-979. [] A. Badawy and C.R. McInnes, "Searation distance for robot motion control using suerquadric obstacle otentials", International Control Conference, aer no 5, lasgow, UK, Setember 6. [] J-C. Latombe, "Robot motion lanning", Kluwer Academic Publishers, 99, Chater 7. [3] A. Badawy and C.R. McInnes, " eneralized Potential Function Aroach for On-Orbit Assembly", IAC 58 th International Astronautical Congress, Hyderabad, India, Setember, 7, submitted for ublication. ISBN:978-988-9867--6 WCE 7