Time-scaling control of a compass type biped robot P Fauteux, P Micheau and P Bourassa Université de Sherbrooke, Mechanical Engineering Departement, Sherbrooke, QC, J1K 2R1, Canada. Email :
[email protected] Phone: (819) 821 8000 ext. 2161 Fax: (819) 821 7163
ABSTRACT This paper presents a robust control strategy driving an actuated compass gait robot towards steady and periodic gaits. By robust we mean a large basin of attraction for the limit cycle. The originality lies in the generation of the swing leg reference angle and speed as a simple function of the supporting leg angle. Simulations and experimentation with a prototype showed that the system exhibits asymptotically stable walking cycles with large and strong basins of attraction.
1
INTRODUTION
Investigations involving biped robots are usually done with one the following goals in mind: creating a complementary tool to clinical studies for the understanding of human walking mechanisms or paving the way for future robotic developments in various emerging fields. Either way, the complexity of the bipedal gait remains an obstacle to its understanding and explains why searchers in the field spend years understanding its subtleties. The compass gait, as described by Goswani et al. [1] and Garcia, Ruina and al [2] is widely accepted as the simplest model of bipedal locomotion and is also recognized, by the biomechanists, as the most basic sub-action that explain the overall walking mechanism. This simplicity allows much insight into the dynamics and control of the human gait and the drawing of strong foundations for the future development of more complex walkers. Furthermore, this gait being dynamic, the kinetic energy is partly transferred from one step to the other, permitting energy efficiency which, according to a general feeling, will be a major obstacle to the implementation of walking robots in the everyday life. McGeer [3] and many others already demonstrated the possibility of near passive gaits. The goal of this study was to create the simplest dynamic biped robot walking on a level ground with a stable and periodic limit cycle and an attractor large and strong enough to recover quickly after a perturbation. Furthermore, it was required that the robot’s behaviour take advantage of the natural interaction of the gravitational field with kinetic and potential energy to give it a smooth, efficient and natural-like walking pattern. This paper presents both simulation and experimentation results of a
1
simple controller ensuring a large basin of attraction in order to reach a dynamically stable walking mode without any complex planning of the trajectory.
2
Dynamic modelling of the compass gait
The chosen model, shown at figure 1, walks with a compass gait similar to the one described by many authors [1-3] with the important distinction that it doesn’t use an incline plane as an energy source. Rather, in order to recover from energy losses due to friction and impacts or to modify the walking cycle frequency, torque is applied from the stance leg to the swing leg and an impulsive force is generated under the post-impact swing leg foot. To avoid unwanted contact with the ground during the swing phase, the robot possesses retractable feet of negligible mass.
T
F Fig. 1. Theoretical model.
The definition of the parameters and variables of the theoretical model are summarized in the table 1. The dynamics of this model may be described by a hybrid set of nonlinear ordinary differential equations and instantaneous algebraic switching relations. These relations, presented in this section, were used by the authors to study, by simulation means, possible control strategies [4-6]. Motion equations developed using the Lagrange formalism and verified trough the Kane formalism are presented in a classic form:
A( q) q + C ( q, q ) + G ( q) = U
(1)
− cos( q1 − q2 ) / 2 where q = qq1 is the vector of angular coordinates, A = I / ml + 1/ 4 the − − cos( q q ) / 2 I /( ml 2 ) + 5 / 4 + me / m 2 1 2 2
inertia matrix; C = − sin( q1 − q2 )q22 / 2 the vector of nonlinear inertial effects (Coriolis, centrifugal), sin( q1 − q2 ) q1 / 2 2
− g cos( q1 ) /(2l ) G = ( me / m + 3 / 2 ) g cos( q2 ) / l
the
vector
of
gravitational
forces,
and
U the vector of external forces. The command of the robot is the T /( ml 2 ) U = 2 −T /( ml ) + F sin( q1 − q 2) /( ml )
torque acting from the stance leg to the swing leg, T. The term F is not commanded, it is an impulsive force (null except for short periods) which models the effect of ground collisions. Ground collisions are considered inelastic and without slip. Hence, the velocity of each link following a collision may be estimated by the law of conservation of momentum which states that the angular momentum of the robot about the impacting foot and the angular momentum of the post-impact swing leg about the hip are conserved. This relation may be written in the contracted form of equation 2 where tf- is the time just prior to the impact and tf+ the instant following:
(2)
q (t +f ) = P q (t −f ) − A−1 J T ( JA−1 J T ) −1 s(t −f )
sin( q2 )q2 + sin( q1 ) q1 the speed of the where P = 10 10 the matrix of permutation of the legs, s = l −cos( q ) q + cos( q ) q
swing foot and J the Jacobian matrix defined as J ij =
3
2
2
1
1
∂si . ∂q j
THE CONTROLLER
3.1 Simplifying hypotheses
To define the controller of the compass gait, some simplifying hypotheses are made. Equation 1 can be rewritten in the form of equation 3. (3)
q = − A−1 ( q)C ( q, q ) − A−1 ( q)G ( q) + A−1 ( q)U
The term A-1(q)U quantifies the effect of the input on the acceleration vector. Substituting the variables by the values of the prototype shows that the torque input as less effect on the stance leg than on the swing leg (at least 31 times for our robot). This ratio, function of the angle from one leg to the other, is minimal when this angle is null. Secondly, the robot evolves within small angles about the vertical. This leads to the conclusion that, conditionally to sufficient kinetic energy, the loss of speed of the supporting leg during a half step is negligible and that its position varies linearly through time. This is verified by experimental results shown at figure 10, where τ is directly proportional to the angle of the stance leg about the vertical. We will therefore make the assumption that the evolution of the stance leg is uncontrollable ( q2 = 0 ) and dependent only on initial conditions at each step:
[
q2 (t ) = q2 (∆t )(t − ∆t ) + q2 (∆t ) for t ∈ ∆t ; t f
]
(4)
For the swing leg is controllable and can be written as J m q1 = T
(5)
with J m the equivalent inertia.
3.2 Time scaling control Figure 2 presents the structure of the control system used to implement the time-scaling control. A supervisor manages the robot with a 6 stages sequential function chart. Stage 1 represents the
impulse under the left foot, stage 2 the control of the left leg, stage 3 the ground impact of the left foot, stage 4 the impulsion under the right foot, stage 5 the control of the right leg, and finally stage 6 the ground impact of the right foot. Thee amplitude and duration the impulsions are constant values; hence, the initial conditions at each step are not commanded. The only one command is the torque T. It is controlled by a sate feedback in order to track the angle and angular speed reference of the hip joint generated according to the parameter A delivered by the supervisor.
left leg impulse
F-left impact
right leg impulse Supervisor
A
Virtual time
τ
F-right
Φ
x ref
θ ref θ ref
xˆ
Observer
State feedback
T
θ Biped robot
Fig. 2. Structure of the control system
The main objective is to achieve stable, robust and strongly attractive limit cycles while maintaining efficient, smooth and natural-like walking patterns. The control of the robot can be defined as a problem of planning optimal trajectories of the legs as a function of time [7]. But, this option needs to perfectly reach some desired initial conditions at each step, or to adapt the optimal trajectory of the measured initial conditions at each step. On the other hand, it is possible to define the trajectory as a function of the geometric evolution of the robot [8]. This is equivalent to a time scaling control [9]. The main originality of this paper lies in the generation of a reference for the swing leg as a monotone function of the stance leg angle instead of a function of time. For this purpose, the function τ, called “virtual time”, proportional to Φ and function of A, the imposed amplitude of a step, is created. The function τ varies, according to our hypotheses, linearly and undisturbed by small torques. It may thus be used to build a reference for the swing leg. A sinusoidal trajectory is expected from a simple pendulum moving within small angles and with a hinge moving with a near constant speed, which is a reasonable assumption according to our hypotheses. An angular velocity reference, with low authority and dedicated to smoothing the swing leg oscillations, is built using the constant ω set to the approximate time derivative of τ. This bypasses the derivation of noisy signals hard to filter without producing destabilizing delays.These relations are presented in equations 6 to 8. A saturation bounds τ between –1 and 1. The duration of a step is imposed only by the dynamics of the stance leg.
(6)
2q 2 (t ) A
τ (t ) = sat Θref (t ) = q2 (t ) +
A 2
sin τ (t )
π 2
ref (t ) = q (t ) + Aπ ω cos τ (t ) π Θ 2 4 2
3
(7)
(8)
PROTOTYPE AND EXPERIMENTAL SETUP
To experiment with this model, a prototype, visible at figure 3, was realized. Parameters of the model are listed in table1. The telescopic legs are driven by three-positions pneumatic cylinders. They generate impulsive forces and avoid unwanted ground collisions. A brushless motor is used to generate the torque acting from the stance leg to the swing leg. The power supply and servo amplifier used allow direct current control and energy accumulation if working against the angular velocity.
Fig. 3. Images of the prototype during conception stage Table 1. Parameters of the model Symbol me m I T F l
Definition Lumped mass of the hip Mass of each leg Mass moment of inertia of each leg around its center of mass Torque acting from the stance leg to the swing leg Impulsive force under the foot of the post-impact swing leg Length of a leg
Parameter value 5.63 Kg 4.02 Kg 0.7 Nm2 -25 to 25 Nm 150 N, 0.04 s 1m
The system is controlled using dSpace rapid prototyping tool and programmed using Matlab/Simulink. Two potentiometers and two switches are the only feedback used so that the angle of each leg and the moment of ground collision might be known. The sampling frequency is 1 kHz. The solver is ODE5.
5
SIMULATION RESULTS
To solve the hybrid algebraic and differential equations describing the robot dynamics, Matlab/Simulink with ODE5 was used. The simulator thus built was used to tests the control strategy. The simulation and experimental model use the same parameters and show strong correlation.
2
2
1.5
1.5
1
1 dqleft/dt (rad/s)
dqleft/dt (rad/s)
The gait stability being one of our main concerns, the first tests were conducted to prove the existence of a limit cycle and its attracting nature. Figure 4 shows a projection of the state phase for one of the legs during a long walk. Figure 5 is similar but with different initial conditions. Such projections show only two of the four state variables and therefore are not preserving all the information. They nonetheless proved to be interesting quick visualization tools.
0.5 0
Impulsions
Convergence
0 -0.5
-0.5 -1 -1.5
0.5
-1
Impacts 1.3
1.4
1.5
1.6 1.7 qleft (rad)
1.8
1.9
Fig. 4. Projection illustrating the limit cycle
2
-1.5
1.3
1.4
1.5
1.6 1.7 qleft (rad)
1.8
1.9
Fig. 5. The attractive nature of the limit cycle
2
A Poincaré map permits a more rigorous visualization. The state space is analyzed every time the stance leg crosses a specific position. The undetermined state variables are thus brought down to three. Figure 6 shows such a map. Data is taken following the impulsions. The robot starts with arbitrary conditions and converges towards a limit cycle defined as the state vector producing an identical state after any number of steps. Figure 7 is a somewhat confusing first visual representation of the basin of attraction. It illustrates the state evolution during the first step with different initial conditions. Figures 8 and 9 are similar but with respectively the initial swing leg speed and position being fixed to the value of the limit cycle. Every data that converges is projected on those planes. The axes are set to the span of the testing. Interestingly, the convergence is dependant almost only on the speed of the stance leg and very little of the state of the swing leg. The support leg imposes its dynamics and therefore has a huge influence. A low speed results in a backward fall and an important speed results in a forward fall since the motor is not able to bring the swing leg forward quickly enough. Also, as demonstrated by Goswani et al. [1], the transition rule, applied to a robot with legs of null moment of inertia, reduces the state phase map to a 2D problem. It becomes necessary only to specify the state of the stance leg to fully determine the motion of the subsequent steps. This implies that the transition rule permits convergence even with important variations of the swing leg state for robots approaching this mass distribution.
-0.9
-0.6
Limit cycle
q2 dot (rad/s)
dq2/dt (rad)
-1 -1.1 -1.2 -1.3 0.5
-0.8 -1 -1.2
1 0
q1 dot (rad/s)
q1 (rad)
Fig. 6. Poincaré section shows convergence
1.25 1.2
q1 (rad)
-0.5
Limit cycle
-0.6
-0.7
-0.8
-0.8
-0.9 -1
-0.9 -1
-1.1
-1.1
-1.2
-1.2
-1.3
-1.3 1.25
1.3 q1 (rad)
1.35
Limit cycle
-0.6
-0.7
q2 dot (rad/s)
q2 dot (rad/s)
-0.5
Fig. 7. Poincaré sections for several trials
-0.5
1.2
1.35 1.3
0
1.25 -1
dq1/dt (rad)
1.4
0.5
1.3
-0.5
1.4
Fig. 8. Projection of Poincaré sections with initial position of the swing leg fixed
-0.5
0
0.5
1
q1 dot (rad/s)
Fig. 9. Projection of Poincaré sections with initial speed of the swing leg fixed
The natural energy dissipation of the collisions with the ground is used to drive the system towards a stable limit cycle. Collisions with greater initial speeds result in higher energy losses creating an attracting limit cycle as have been demonstrated in simulations and experimentally.
6
EXPERIMENTAL RESULTS
Sequential Function Chart Stage
Figure 11 shows the evolution of important parameters illustrating the cyclic nature of the walk: τ that is used to build the angular references, stages of the sequential function chart and the in-between leg angle and speed and their references. 1
τ (rad)
0.5 0 -0.5 -1 0
2
4
6
Time (s)
θ θref
4 2 0
0
2
4
0.5 0
6
Time (s)
θ dot θref dot
4
θ dot (rad)
θ (rad)
1
6
2 0 -2
-0.5 0
2
4
6
0
Time (s)
2
4
6
Time (s)
Fig. 10. Important parameters during a walk
Figure 12 plots a Poincaré section similar to those above. The dots represent measures taken during a normal walk. The circles pairs are taken during the two first steps of starting sequences where the robot starts itself with a variable impulsion. The figure shows the existences of an attractor towards which are converging points inside an area. Squares are similar measures, but do not converge. -0.4 -0.5
-0.4
-0.6 -0.7
-0.8
q2 dot (rad/s)
q2 dot (rad/s)
-0.6
-1
-1.2
-0.8 -0.9 -1 -1.1 -1.2
-1.4 1
2
0.5 1.9
0 q1 dot (rad/s)
-0.5
1.8
q1 (rad)
-1.3 -1.4 -0.5
0 0.5 q1 dot (rad/s)
1
Fig. 11. Poincaré section revealing an attractor
We further tested the stability of the robot by placing free obstacles in its path; the robot would hit the object at mid-stance and then struggle to recover its steady gait. The applied perturbations are tripping impulses that reduce the velocity of each link in a ratio that is dependant on the respective masses and inertia. Figure 13 shows a projection of the state phase of one leg during such an event. This perturbation is also visible on figure 14 showing the leg angle its reference. The swing leg impacts the ground before reaching the imposed step length. The limit cycle is reached during the following step.
2.5
θ 2 1.5
θref
0.4
1
Impact
0.5
θ (rad)
qleft dot (rad/sec)
Impact
0.6
0
0.2 0
-0.5
-0.2
-1
-0.4
-1.5
1
1.2
1.4
1.6
1.8
qleft (rad)
Fig. 12. State phase projection with tripping impulse
2
0
0.5
1
1.5
2 Time (s)
2.5
3
3.5
4
Fig. 13. Leg angle and its reference with tripping impulse
7 CONCLUSION The paper presented a possible approach for the control of an actuated walking model inspired by the well known passive compass gait walker. In order to reach a dynamically stable walking mode, a simple control strategy involving passivity mimicking in-between leg angle references built via a function of the geometric evolution of the robot was used. Simulations and experimentation showed the robustness of this strategy. In a more general way, the “virtual time” or geometry based control approach proved its simplicity and applicability to systems with one or more uncontrollable degrees of freedom including the bipedal gait.
REFERENCES 1. Goswami A, Tuillot B, Espiau B (1996) Compass like bipedal robot part 1: Stability and bifurcation of passive gaits. INRIA Research report 2996 2. Garcia M, Chatterjee A, Ruina A, Coleman M (1998) The simplest walking model: stability, complexity, and scaling. J Biomech Eng 120(2):281-8 3. McGeer T (1990) Passive dynamic walking. Int J Robot Res 9(2):62-82 4. Micheau P, Buaka P, Bourassa P (2002) Control of the simplest walking model with Lambda model. Automatic Control 15th IFAC World Congress Barcelona, 6 pages 5. Micheau P, Roux M, Bourassa P (2003) Self-tuned trajectory control of a biped walking robot. clawar, 527-534 6. Bourassa P, Micheau P (2002) Tripping impulses – gait limit cycle for biped. clawar. 791-798 7 Qiang H, Kazuhito Y, Shuuji K, Kenji K, Hirohiko A, Noriho K, Kazuo T (2001), Planning walking patterns for a biped robot, IEEE Transactions on robotics and automation, 17(3), 280-289 8 Chevallerau C, (2003) Time-scaling control for an underactuated biped robot. IEEE trans. robotics and automation. vol. 19, no. 2, 363-368. April 2003 9 Dahl O, Nielsen L(1990), Torque-limited path following by online trajectory time scaling, IEEE Trans. Robot. Automa., vol. 6, pp. 554-561 10 Web site of P.Micheau, http://mecano.gme.usherbrooke.ca/~pmicheau/
