Full paper Minimizing Energy Consumption in Hexapod Robots

Advanced Robotics 23 (2009) 681–704www.brill.nl/arFull paperMinimizing Energy Consumption in Hexapod RobotsP. Gonzalez de Santos ∗ , E. Garcia, R. Ponticelli and M. ArmadaInstitute of Industrial Automation — CSIC, Department of Automatic Control, Ctra. Campo RealKm 0,2, 28500 Arganda del Rey, Madrid, SpainReceived 15 July 2008; revised 25 November 2008; accepted 3 December 2008AbstractMinimization of energy expenditure in autonomous mobile robots for industrial and service applications isa topic of huge importance, as it is the best way of lengthening mission time without modifying the powersupply. This paper presents a method to minimize the energy consumption of a hexapod robot on irregularterrain. An energy-consumption model is derived for statically stable gaits before applying minimizationcriteria. Then, some parameters that define the foot trajectories of the legged robot are computed to minimizeenergy expenditure during every half a locomotion cycle. The method is evaluated using an accurategeometric model of the SILO-6 walking robot.© Koninklijke Brill NV, Leiden and The Robotics Society of Japan, 2009KeywordsEnergy efficiency, energy consumption, legged locomotion, mobile robots, optimization methods1. IntroductionAn autonomous mobile robot is an intelligent machine with the skills to performmotion in an unstructured environment without explicit human intervention. Itsintelligent autonomy is provided by efficient algorithms capable of coping withmany different situations. Nevertheless, a real-application autonomous robot mustalso exhibit energy autonomy; the greater the autonomy, the better. Both sorts ofautonomies are required for real operative mobile robots. Intelligent autonomy isprovided by sensors and algorithms. Energy autonomy is mainly accomplished usingefficient power supplies. However, the control algorithms that mobile robots usecan help lengthen the duty cycle of power supplies and so extend the robot’s operationtime. This is especially significant in legged locomotion (both multi-legged andbiped robots), which can move along a given path with many different leg and bodyposes and, thus, with many different energy expenditure rates. Even though multi-* To whom correspondence should be addressed. E-mail: pgds@iai.csic.es© Koninklijke Brill NV, Leiden and The Robotics Society of Japan, 2009 DOI:10.1163/156855309X431677

P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704 683Figure 1. SILO-6 walking robot.Thus, we can see the importance elevation maps have held for autonomous mobilerobots on irregular terrain for about the last decade. Elevation maps have beenused mainly for navigation purposes in general, and, in some legged-robot applications,elevation maps have also provided a starting point for finding adequatefootholds to improve traction and reduce foot slippage. However, terrain informationcan also help reduce the support torques of legged robots and, thus, reduce theenergy required.This paper focuses on the minimization of energy expenditure in hexapod robotswalking on irregular terrain based on aprioriknowledge of the terrain in front ofthe robot. This work is motivated by a real need to increase the energy autonomyof the SILO-6 walking robot. The SILO-6 is a six-legged robot developed as a mobileplatform for the DYLEMA project [15, 16], whose main aim is to develop alocomotion system to integrate relevant technologies in the fields of legged robotsand sensors to address needs in humanitarian demining activities. The hexapod isendowed with a 5-d.o.f. manipulator in front, which handles a sensor head to scanthe terrain in the front of the hexapod covering its whole sprawl (see Fig. 1). Thisscanning manipulator is fast enough to scan the terrain without jeopardizing thehexapod speed. The sensor head consists of a mine-detecting set and a network ofinfrared (IR) sensors. The detecting set is a commercial device capable of detectingvery small metallic components. The IR sensors are used to maintain the sensorhead at a given height above the terrain and roughly parallel to any terrain irregularities.Thus, this sensor head can be used to generate an elevation map of whatlies in front of the robot, which can in turn be used to minimize the energy requiredto move the robot. Figure 2 shows an example of reconstruction of an irregular terraincomposed of different geometric objects using the described sensor head and a

P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704 689These foot forces depend on the foot positions that determine matrices R i ,i ∈[p,q,r], andR R , and in this case they are defined by the alternating-tripod gait(see Refs [21, 23] for foothold computation). Note that the wrench W pqr in (21)contains the inertial forces and moments, and the external forces as well. Inertialforces and moments are caused by the motion of the masses of the leg links. Theseterms are obtained by computing first the accelerations of the COG of all of thelinks, COG L , along a semi-cycle. These accelerations are determined by the accelerationsof the feet that will be defined below. The only external force acting in therobot is the weight, W , taking into consideration the condition C-2.The solution of (21) along with (10) provides the joint torques. The speed inevery joint is given by:ω ∗ i = J−1 iv i , (22)where ω ∗ i and v i are given by:ω ∗ i = (ω∗ i1ωi2 ∗ ωi3 ∗ )T = ( ˙θi1∗ ˙θi2∗ ˙θi3 ∗ )T (23)v i = ( ẋ i ẏ i ż i ) T .θ ij is the position of joint j of leg i and (x i y i z i ) T is the position of foot i.Again, ωij ∗ represents the speed in joint j of leg i, while ω ij represents the speed inmotor j of leg i.The speed and torques in joint j and motor j of leg i are related through thejoint-gear ratio, N j , and the joint gear’s mechanical efficiency, η j , by:ωij ∗ = ω ijN j(24)τij ∗ = η j N j τ ij .Therefore, (8) can be re-written as:∫ T 6∑ 3∑( ( ω∗ij(t)τij ∗E =(t) ))R j+0η j k 2 i=1 j=1M jηj 2N j2 (τij ∗ )2 (t) dt. (25)Thus, the algorithm to compute the energy consumed in a hexapod consists of:Step 1. Set the body velocity, body height, leg stroke R x and foot trajectories.Step 2. Compute foot positions (x i (k)y i (k)z i (k)) T for instant k using thealternating-tripod gait [21, 23].Step 3. Compute the trajectory, speed and acceleration of every COG L .Step 4. With both the COG L accelerations and link masses compute the inertialforces and moments.Step 5. Compute foot forces f i by solving the force-distribution problem with (21).Step 6. Compute torques τij ∗ using (10).

690 P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704Step 7. Compute joint speed ωi ∗ with (22).Step 8. Compute the total energy during half a locomotion cycle with (25), whereT = T cycle /2.4.2. Kinematic Model of the HexapodSection 1 mentioned that the main aim of this work is to configure an energyefficientwalking robot for humanitarian demining activities. This robot model isconsidered herein for simulation purposes. The mechanical and geometric parametersof SILO-6 [15, 16] are defined in Fig. 4 and summarized in Table 1. With thoseparameters, a wire model of the robot can be obtained as illustrated in Fig. 5. Theasterisk represents the location of the COG L of each link. The COG of the body,COG B , is assumed to be at the geometric center of the body. With this model, wecompute the position of the overall COG of the robot, COG R , as well as the forcesand moments acting on it, which depend on the motion of the feet.The foot positions have been designed to minimize the accelerations and, thus,to reduce the dynamic effects. The foot trajectory for each foot in an external referenceframe will consists of a straight trajectory for the foot in its support phase anda transfer trajectory composed of three subtrajectories: a circular trajectory to risethe foot, a straight trajectory to move the foot forward and a circular trajectory toTable 1.Main SILO-6 featuresBodydimensions (m)length L B 0.88front/rear width D 0.38middle width 0.63height 0.26stroke pitch P x 0.365mass (kg) 28.2speed (m/s) 0.05Leglinklength(m)a 1 0.084a 2 0.250a 3 0.250stroke (R x ) (m) 0.25mass (kg) 4.3foot speed (m/s)transfer phase 0.140support phase 0.05Robottotal mass (kg) 54See Fig. 4 for parameter definitions.

P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704 691Figure 5. Wire model of the hexapod. Foot positions are indicated in dark circles. The COG of everylink is indicated with an asterisk.place the foot on the ground (Fig. 6b). With these trajectories, we guarantee that footaccelerations at foot takeoff and landing are minimized. Figure 6 plots the foot trajectoryin both the body reference frame (Fig. 6a) and the external reference frame(Fig. 6b). Note that the foot trajectory in the body reference frame is transformedto joint trajectories by using the leg kinematic model described in the Appendix.Applying these joint trajectories into the hexapod’s controller (see a superpositionof robot poses in Fig. 7), the inertial forces and moments acting in the COG R arecomputed.It is worth clarifying that of the energy expended in the walking mechanism(stand, traction, leg swing and head loss), the contribution of the leg swing is insignificant.Every leg in its stand phase must support about one-third of the robot’sweight, while a leg in the swing phase does not support any load. That means thelegs in the swing phase do not expend energy in comparison to the legs in the standphase. Moreover, all the legs in the swing phase expend quite the same energy;therefore, the difference between two foothold solutions is basically the differencebetween their support phases. From the theoretical standpoint, there is, of course,a slight difference in the energy consumed for placing a foot in different points atthe end of the swing phase, but they are quite close to each other and a swing phaseis an unload motion, so this energy can be neglected. Additionally, the algorithmrelies on an optimization method and the model must be kept significant enough,but yet small to be computationally efficient. This is the reason that in our modelwe discard the energy consumed in the swing legs.

692 P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704(a)(b)Figure 6. Foot trajectories seen by an observer at (a) the body reference frame (trajectory in supportfrom A to B) and (b) the external reference frame (C and D are the support points).Figure 7. Superposition of robot poses showing the foot and COG L trajectories.

4.3. Simulation ResultsP. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704 693Let us compute the energy expended by the SILO-6 model when the robot walksalong a straight, horizontal line with its body leveled and performing the foot trajectoryplotted in Fig. 6 (see also Fig. 7). The body speed is assumed to be 0.1 m/s,which means the feet in support move at 0.1 m/s; however, the feet in the transferphase reach up to 0.55 m/s; the leg stroke of the gait, R x ,is0.25mandthedistance from every foot trajectory to the z-axis of its leg reference frame, L i ,is0.25 m for feet in the support phase (tripod i ∈[1, 4, 5] in the half locomotion cycleunder study). With these premises and the mechanical, geometric and electrical parametersprovided in Tables 1 and 2, we obtain the motor torques and motor speedduring half a locomotion cycle plotted in Fig. 8. These magnitudes are the base toobtain the power. For the sake of brevity, only the torques and speed for leg 1 areillustrated. Note that for an ideal mass-less leg robot the torque in motor 1 shouldTable 2.SILO-6 joint parametersLeg joint Motor GearTorque constant Resistance Rate EfficiencyK M (Nm/A) R () N η (%)1 0.0333 0.62 246 632 0.0333 0.62 881.5 423 0.0333 0.62 881.5 42Figure 8. Torques and speed in motors of leg 1 during half a locomotion cycle.

694 P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704Figure 9. Mechanical power in motors.be null, because motor 1 does not contribute to robot support or motion at constantspeed; however, Fig. 8 shows that in that joint the torque is not completely null becauseof the effect of the COG L of the links. Note that in this case the robot is onflat, horizontal terrain.Figure 9 plots the mechanical power consumption, along half a locomotion cycle,while legs 1, 4 and 5 are in support. Figure 10 plots the heat loss for the samelegs and same conditions. These power contributions have been plotted individuallyfor the sake of comparison. We can observe that heat loss is greater than mechanicalpower and the area under the heat loss (energy) is also greater than the areaunder the mechanical power. This fact highlights the importance of heat loss overmechanical power — a term that has been neglected in many studies. For instance,the pioneering work of Lapshin [2] considers only the energy consumption on themachine weight, the energy consumption on the traction generation and the energyfor the leg swing phases. A more recent work [24] minimizes the sum of the squareacceleration through the entire trajectory. However, by adding the two power contributionsand integrating this function over half a locomotion cycle, we find thatthe mechanical energy required to support and move our hexapod is E M = 17.22 J,

P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704 695Figure 10. Heat loss in motors.while the energy due to heat loss is E H = 37.01 J. That makes a total energy ofE T = 54.23 J for a tripod (half a locomotion cycle), which means that the heat lossis about 68.24% of the total energy.5. Minimization of Energy ConsumptionEnergy consumption depends on motor torque and motor speed, and these parametersdepend on foot positions, as mentioned above. Therefore, varying the foottrajectories is one way of changing the energy required to support and propel arobot.In our approach, the leg stroke, R x , and leg phases (i.e., the leg-lifting instantsin a locomotion cycle) remain constant. The period of the locomotion cycle willdepend on the body speed. Foot trajectories in support are straight segments of afixed length (R x ); thus, the only parameter for defining different foot trajectoriesis the distance from the foot trajectory to the z-axis of the leg reference frame, L i ,and the foot trajectory height, H i (see Fig. 4). Note that the robot is regarded asmoving in a horizontal straight line with respect to a fixed external reference frame.

P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704 697constrained, nonlinear minimization method also known as the flexible polyhedronmethod [25]. This method requires a starting point (L p0 ,L q0 ,L r0 ) and finds a localminimum (L ∗ p ,L∗ q ,L∗ r ) of the function E:minimize Lp ,L q ,L r{E(L p ,L q ,L r )}→(L ∗ p ,L∗ q ,L∗ r ). (28)Equation (25) is a unimodal function for the range of input values L i ∈[−0.1 m, −0.35 m] as numerically shown below (Fig. 11); therefore, the minimumfound by (28) can be considered a global minimum. Note that the initial values ofL i are chosen to be inside the leg workspace so that the probability of finding asolution out of the leg workspace is negligible. However, if the inverse-kinematicfunction of the leg finds a point out of the workspace, it is not considered and therobot will perform the next step under non-optimal conditions.5.2. Computing ResultsTo illustrate the algorithm results, let us consider that SILO-6 is walking on flatterrain at a height of H =−0.15 m, and apply (28) to compute the values of L p ,L q and L r that minimize the energy. For the SILO-6 parameters summarized inTables 1 and 2, and starting at the initial solution L p0 = L q0 = L r0 = 0.25 m,the optimal solution is L p = 0.255 m, L q = 0.254 m and L r = 0.255 m, and theminimum energy is 49.63 J.Let us now assume that the robot is walking on an irregular terrain that consistsof a horizontal base plane with several protrusions, P , such that the first tripod hasfootholds on the protrusions at H i =−0.15 m and the second tripod has footholdson the base plane at H i =−0.30 m (Fig. 12). If the robot walks during the first semicyclewith foot trajectories at L p = 0.255 m, L q = 0.254 m and L r = 0.255 m, therobot consumes the minimum energy as shown above. However, if the robot keepsthe same foot-trajectory parameters for the second tripod, the energy expenditureis 52.04 J, whereas reapplication of the minimization algorithm at this point wouldreduce the energy to 50.42 J for L p = 0.210 m, L q = 0.224 m and L r = 0.210 m.Note that the savings of 1.62 J is for a body motion of about R x = 0.25 m at a bodyFigure 12. Sketch of the hexapod over irregular terrain.

698 P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704Table 3.Energy expenditure for different foot positionsH i (m)−0.10 −0.15 −0.20 −0.25 −0.30 −0.35L ∗ 1 (m) 0.248 0.255 0.250 0.235 0.210 0.074L ∗ 4 (m) 0.244 0.254 0.252 0.242 0.224 0.197L ∗ 5 (m) 0.248 0.255 0.250 0.235 0.210 0.074Energy (J)H i ′ −0.10 39.66 39.88 39.76 39.94 42.83 74.60−0.15 49.82 49.63 49.66 50.35 53.46 70.60−0.20 54.27 54.23 54.21 54.54 56.50 64.95−0.25 54.36 54.65 54.49 54.23 54.88 58.13−0.30 51.40 52.04 51.75 50.91 50.42 50.42−0.35 46.19 47.12 46.73 45.42 44.04 41.60speed of 0.1 m/s. These values are shown in Table 3, which includes the energyexpenditure under different body height, H i , and leg spread, L i .InTable3,L ∗ iisthe foot position that minimizes the energy (in bold) for a given H i . H ′i representsthe height of the body at the computation of the energy. The rows in the energypart of Table 3 contain the energy expenditure for a given body height, Hi ′,whenthe foot positions are those indicated in the same column. Thus, the minimum inevery row appears at the cell H i = Hi ′ (indicated in bold). Note that for both supporttripods the body is on the same horizontal plane; therefore, there is not any energyexpended for moving up or down the robot’s body.Figure 13 illustrates the energy surface as a function of H i and Hi ′ . The points ofminimum energy for a given Hi ′ are indicated with an asterisk. The surface showshow the minimum energy expenditure is obtained for low or high body heights,which correspond with the insect or mammal robot configuration. However, for agiven body height, Hi ′ , the minimum energy is obtained at a specific foot positions,L i .Similar examples can be run considering other type of irregular terrain. For instance,the first tripod could be at the base plane with H i =−0.15 m and the secondtripod could be inside holes such as H i =−0.30 m. The results in both examples arethe same. A general irregular terrain with the footholds at different heights can alsobe used, but we prefer to present a simple irregular terrain for the sake of clarity.6. Practical Implementation IssuesThe earlier sections have presented the theory for minimizing the energy consumptionin a hexapod robot when it performs a tripod gait along a straight line.

P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704 699Figure 13. Energy expenditure for different body heights.A numerical example of the procedure has illustrated that it is possible to saveabout 3.21% of the energy for a hypothetical, special case where every foot in atripod is at the same height. In such a case, a lookup table with the solution to theproblem could be an adequate practical solution to avoid computing the optimizationprocedure in real-time.In a practical case, when the robot is walking on irregular terrain every footwill normally be placed at a different height in a range from −0.10 m up to about−0.35 m. The lookup table solution seems to be unpractical, even if we use a reasonablediscretization of the foot heights, and the implementation in real-time ofthe minimization algorithm is the only sensible solution, which brings about furthertechnical computing problems.Let us assume that we know, as in our real application introduced in Section 1,an elevation map of the terrain where the robot is stepping on. That means we knowthe z-foothold component as a function of the x and y foothold components. Thus,we can compute the minimization algorithm at the beginning of the transfer phase,which lasts for about t T = 2.5 s, taking into consideration the average speed ofthe transfer phase v T = 0.1 m/s and the leg stroke R x = 0.25 m. This time t T isinsufficient for running the minimization algorithm in the PC-bus based computer(industrial ISA PC Pentium II 667 MHz) on board SILO-6. Additional computingresources must be provided on the prototype to implement the method presentedin this paper. This is part of the technical work envisaged for the next step of the

700 P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704project. Nevertheless, the most important result is to quantify the range of energyconsumption we can save by applying minimization methods.7. ConclusionsEnergy optimization is an important topic for autonomous robots, because it is theway of increasing duty time for missions without modifying the power supply onboard the robot.This paper studies the energy required for six-legged robots using alternatingtripodgaits, the fastest gait for hexapods, and derives a method to minimize theenergy for one tripod. Minimization of consecutive tripods minimizes the energythroughout an entire trajectory. The method works on any kind of irregular terrain,although some simple models for irregular terrain have been used in the examplespresented above.The method requires some advance knowledge of the terrain; therefore, it isapplicable to robots capable of obtaining an elevation map of the terrain. Energyexpenditure was computed by using the SILO-6 walking robot as a model. This robotis designed for humanitarian demining missions and uses a system to obtain anelevation map of the terrain in front of the robot.This paper provides some figures for the energy saved in such a robot and researchis ongoing on the computation of the method for real-time applications.Energy consumed by electronics (computers, drivers, analog I/O, etc.) constitutes alarge source of energy loss. Diminishing this energy loss is a technical concern thatmust always be taken into account.AcknowledgementsThis work was supported by the Spanish Ministry of Education and Science throughCICYT grant DPI2004-05824.References1. Q. Huang, T. Hase and K. Ono, Optimal trajectory planning method using inequality state constraintfor biped walking robot with upper body mass, Special Issue on New Trends of Motion andVibration Control, J. Syst. Des. Dyn. (JSME Dyn. Meas. Control Div.) 1, 168–179 (2007).2. V. V. Lapshin, Energy consumption of a walking machine. Model estimations and optimization,in: Proc. 7th Conf. on Advanced Robotics, San Feliu de Guixols, pp. 420–425 (1995).3. D. E. Orin and S. Y. Oh, Control of force distribution in robotic mechanisms containing closekinematic chains, J. Dyn. Syst. Meas. Control 103, 134–141 (1981).4. M. A. Nahon and J. Angeles, Minimization of power losses in cooperating manipulators, J. Dyn.Syst. Meas. Control 114, 213–219 (1992).5. D. W. Marhefka, Gait planning for power minimization in walking machines, Master’s Thesis,Ohio State University (1996).6. D. W. Marhefka and D. E. Orin, Gait planning for energy efficiency in walking machines, in: Proc.IEEE Int. Conf. on Robotics and Automation, Alburquerque, NM, pp. 474–480 (1997).

P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704 7017. M. F. Silva, J. A. Tenreiro Machado and A. M. Endes Lopes, Energy analysis of multi-leggedlocomotion systems, in: Proc. 4th Conf. on Climbing and Walking Robots, Karlsruhe, pp. 143–150(2001).8. T. Zelinska, Efficiency analysis in the design of walking machines, J. Theor. Appl. Mech. 38,693–708 (2000).9. V. V. Zhoga, Computation of walking robots movement energy expenditure, in: Proc. IEEE Int.Conf. on Robotics and Automation, Leuven, pp. 163–168 (1998).10. J. E. Bares and D. S. Wettergreen, Dante II: technical description, results and lesson learned, Int.J. Robotics Res. 18, 621–649 (1999).11. J. Bares and W. Whittaker, Configuration of autonomous walkers for extreme terrain, Int. J. RoboticsRes. 12, 535–559 (1993).12. Nomad: Lunar Rover Navigation, http://www.cs.cmu.edu/∼lri/nav97.html (1997).13. C. Ye and J. Borenstein, A new terrain mapping method for mobile robot obstacle negotiation, in:Proc. UGV Technology Conf. at the 2003 SPIE AeroSense Symp., San Jose, CA, pp. 2561–2562(2003).14. P. Pfaff and W. Burgard, An efficient extension of elevation maps for outdoor terrain mapping, in:Proc. 5th Int. Conf. on Field and Service Robotics, Port Douglas, QLD, pp. 165–176 (2005).15. P. Gonzalez de Santos, J. A. Cobano, E. Garcia, J. Estremera and M. A. Armada, A six-leggedrobot-based system for humanitarian demining missions, Mechatronics 17, 417–430 (2007).16. SILO-6 website, http://www.iai.csic.es/users/silo617. S. M. Song and K. J. Waldron, Machines that Walk: the Adaptive Suspension Vehicle. MIT Press,Cambridge, MA (1989).18. M. Kaneko, S. Tachi, K. Tanie and M. Abe, Basic study on similarity in walking machines from apoint of energetic efficiency, IEEE J. Robotics Automat. 3, 19–30 (1987).19. J. Nishii, K. Ogawa and R. Suzuki, The optimal gait pattern in hexapods based on energetic efficiency,in: Proc. 3rd Int. Symp. on Artificial Life and Robotics, Beppu, Vol. 1, pp. 106–109 (1998).20. S. Ma, Time optimal control of manipulators with limit heat characteristics of actuator, Adv. Robotics16, 309–324 (2002).21. P. Gonzalez de Santos, E. Garcia and J. Estremera, Improving walking-robot performances byoptimizing leg distribution, Autonomous Robots 23, 247–258 (2007).22. W. Y. Jiang, A. M. Liu and D. Howard, Foot-force distribution in legged robots, in: Proc. 4th Int.Conf. on Climbing and Walking Robots, Karlsruhe, pp. 331–338 (2001).23. P. Gonzalez de Santos, J. Estremera and E. Garcia, Optimizing leg distribution around the body inwalking robots, in: Proc. IEEE Int. Conf. on Robotics and Automation, Barcelona, pp. 3218–3223(2005).24. R. Kurame, K. Yoneda and S. Hirose, Feedforward and feedback trot gait control for quadrupedwalking vehicle, Autonomous Robots 12, 157–172 (2002).25. J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder–Mead simplex method in low dimensions, SIAM J. Optimizat. 9, 112–147 (1998).AppendixThis Appendix presents the leg kinematics used for simulation and control purposes(Fig. A.1). The direct kinematic relationships consist in envisioning the position(x i ,y i ,z i ) T of foot i as a function of the joint components (θ i1 ,θ i2 ,θ i3 ) T :(x i ,y i ,z i ) T = T DIR (θ i1 ,θ i2 ,θ i3 ), (A.1)

702 P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704Figure A.1. SILO-6 prototype, dimensions and reference systems.where⎛⎞Cθ i1 (a 3 C(θ i2 + θ i3 ) + a 2 Cθ i2 + a 1 )T DIR = ⎝ Sθ i1 (a 3 C(θ i2 + θ i3 ) + a 2 Cθ i2 + a 1 ) ⎠ .a 3 S(θ i2 + θ i3 ) + a 2 Sθ i2Note that we simplify the notation by writing C for cos and S for sin.The inverse kinematics is defined by:where:and:(A.2)(θ i1 θ i2 θ i3 ) T = T INV (x i ,y i ,z i ) for i = p,q,r, (A.3)⎛T INV =⎜⎝( )yiarctgx( i( )Bi− arctg + arctgA i(arctgD i√± A 2 i + B2 i − D2 i)z i − a 2 Sθ i2x i Cθ i1 + y i Sθ i1 − a 2 Cθ i2 − a 1⎞), (A.4)⎟⎠− θ i2A i =−z iB i = a 1 − x i Cθ i1 + y i Sθ i1(A.5)D i = 2a 1(x i Cθ i1 + y i Sθ i1 ) + a 2 3 − a2 2 − a2 1 − x2 i − y2 i − z2 i2a 2.

P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704 703Finally, the Jacobian matrix is:( −Sθ1 (a 3 C(θ 2 + θ 3 ) + a 2 Cθ 2 + a 1 )J = Cθ 1 (a 3 C(θ 2 + θ 3 ) + a 2 Cθ 2 + a 1 )0−Cθ 1 (a 3 S(θ 2 + θ 3 ) + a 2 Sθ 2 ))−a 3 Cθ 1 S(θ 2 + θ 3 )−Sθ 1 (a 3 S(θ 2 + θ 3 ) + a 2 Sθ 2 ) −a 3 Sθ 1 S(θ 2 + θ 3 ) . (A.6)a 3 C(θ 2 + θ 3 ) + a 2 Cθ 2 a 3 C(θ 2 + θ 3 )Joint reference frames and leg parameters are defined in Fig. A.1. Link lengthvalues, a i , are indicated in Table 1.About the AuthorsPabloGonzalezdeSantosis a research scientist at the Spanish National ResearchCouncil (CSIC). He received his PhD degree from the University of Valladolid,Spain. Since 1981, he has been involved actively in the design and developmentof industrial robots and also in special robotic systems. His work during last 15years has been focused on walking machines. He worked on the AMBLER projectas a Visiting Scientist at the Robotics Institute of Carnegie Mellon University.Since then, he has been leading the development of several walking robots suchas the RIMHO robot designed for intervention on hazardous environments, theROWER walking machine developed for welding tasks in ship erection processes, and the SILO-4walking robot intended for educational and basic research purposes. He is now leading the DYLEMAproject, which includes the construction of the SILO-6 walking robot, to study how to apply walkingmachines to the field of humanitarian demining. He has published 45 articles in scientific journals andone textbook on quadrupedal locomotion.Elena Garcia is a Researcher at the Spanish National Research Council (CSIC).She received the BE and PhD degrees from the Universidad Politécnica de Madrid,in 1996 and 2002, respectively. She has been a Visiting Student at the MassachusettsInstitute of Technology, in 1998, and at the Laboratoire d’Automatiquede Grenoble, in 2001. She has participated in various research projects on fieldand service legged robots like ROWER, a walking platform for ship building,SILO-4, a four-legged locomotion system used as a testbed in most of her work,and DYLEMA, a project aimed at using walking robots for antipersonnel mine detectionand location. Her research topics include dynamic stability of walking robots, gait adaptationto environmental disturbances, active compliance in walking robots and design of large specific-poweractuators. She has published 20 articles in SCI scientific journals, 25 conference proceedings and onescientific book.Roberto Ponticelli is a PhD student at the Complutense University, Madrid,Spain. He received his Electronic Engineer degree from the Simon Bolivar University,Caracas, Venezuela, in 1998, and since then he has been involved in thedesign and development of animatronics robots and industrial automation systems.Since 2003, he has been collaborating with the Automatic Control Department ofthe Institute of Industrial Automation of the Spanish National Research Council(DCA-IAI-CSIC), and he is now participating in the DYLEMA project, thatincludes the construction of the SILO-6 walking robot, to study how to applywalking machines to the field of humanitarian demining.

704 P. Gonzalez de Santos et al. / Advanced Robotics 23 (2009) 681–704Manuel A. Armada received the PhD degree in Physics from the University ofValladolid, Spain, in 1979. Since 1976, he has been involved in research activitiesrelated to control theory (singular perturbations and aggregation, bilinear systems,adaptive and nonlinear control, multivariable systems in the frequency domain,and digital control) and robotics (teleoperation, climbing and walking robots). Hehas been working in more than 50 RTD projects (including international ones likeEUREKA, ESPRIT, BRITE/EURAM, GROWTH, and others abroad the EU, especiallywith Latin-America (CYTED) and Russia. He owns several patents andhas published over 150 papers (including contributions to several books, monographs, journals, internationalcongresses and workshops). He is presently Vice-director of the Instituto de AutomaticaIndustrial (IAI-CSIC), Member of the Russian Academy of Natural Sciences, and Doctor HonorisCausa by the State Technical University (MADI) of Moscow. His main research direction is concentratedin robot design and control, with special emphasis in fields like force control and walking andclimbing machines. He has been presented with the IMEKO TC17 Award and with the CSIC Award.