Dynamic Modeling and Numerical Simulations of a Passive Robotic Walker Using Euler- Lagrange Method Irfan Hussain1*, IEEE Member, PhD, Mohammad I. Awad1, IEEE Student Member, Ali Bin Junaid3 Federico Renda1,2, IEEE Member, PhD, Lakmal Seneviratne1, IEEE Member, PhD, Dongming Gan1, PhD
Abstract— The two main important factors to be considered while designing an efficient biped robot are human like gait cycle and energy efficiency. The researchers have sought the solution through passive dynamic walking which can provide highly energy efficient model for biped locomotion exhibiting human like gait under gravity. By adding actuation at some joints, the passive dynamic walking robot can walk stably on level ground. In this paper, we present numerical simulation of a simple robotic dynamic walker using Matlab. We derived the equation of motion using Euler-Lagrange method. We first demonstrate that a simple passive biped walker, vaguely resembling human legs, can walk down a shallow slope without any actuation and control. Later, we extended the passivity property to active dynamic walkers on level ground. Our model is the simplest special case of the passive-dynamic models pioneered by McGeer (1990a). It has two rigid massless legs hinged at the hip, a pointmass at the hip, and point-masses at the feet. After nondimensionalizing the governing equations, the model has only one free design parameter, the ramp slope γ. This model shows stable walking modes similar to more elaborate models, but allows some use of analytic methods to study its dynamics. Keywords—Passive walkers, Biped Robots, Dynamics, Passive dynamic walking
I. INTRODUCTION In the past few decades, robotics research has made huge progress in the area of biped locomotion for various reasons, running from prosthesis development [1], rehabilitation [2], transportation underwater locomotion [3] and entertainment industries [4]. Based on their actuation system, biped walkers are categorized to fully-actuated, under-actuated and passive walkers. Fig. 1 shows an example of each category. Fully actuated biped walkers are widely illustrated in literature [5, 6], this type suffers from the high costs of both energy consumption and control [8]. In order to reduce these costs, the under-actuated walkers were presented (see, Fig. 1, middle). Adding actuation to passive dynamic walkers result in highly efficient robotic walkers. Such walkers can be Irfan Hussain is corresponding author (
[email protected]) 1 Authors are with the Khalifa University Robotics Institute, Khalifa University of Science Technology and Research, Abu Dhabi, United Arab Emirates 2Authors are with the Dept of Mechanical Engineering, Khalifa University of Science Technology and Research, Abu Dhabi, United Arab Emirates
implemented at lower mass and use less energy because they walk effectively with only a couple of motors. Several designs
Figure. 1: From left to right, examples of passive walker, underactuated and fully actuated powered biped robots are shown .
and control laws were developed for this type of biped walkers [7 -9]. A solution for energy efficiency is the exploitation of the ‘natural dynamics’ of the multi-body system, or by incorporating actuators that are energy dissipative, energyredirecting, or energy storing to the system [10]. Recently, some other solutions for exploiting the passive compliance in the underactuated robotic devices have also been introduced [11-13]. In 1989 McGeer [14] introduced the idea of ‘passive dynamic walking’, inspired by the research of Mochon and McMahon [12]. They showed that in human locomotion the motion of the swing leg is merely a result of gravity acting on an unactuated double pendulum. McGeer [16] extended the idea and showed that a completely unactuated and therefore uncontrolled robot can perform a stable walk. Garcia et al [17] have researched several stability and efficiency issues of those passive dynamic walkers, and showed that in the limiting case, energy consumption can even be reduced to zero, like an ideal rolling wheel. The walking 3Department of Mechanical Engineering, KU Leuven, 3000 Leuven, Belgium.
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motion of a passive dynamic walker is started by launching the robot with such initial values for the leg angles and velocities, that the end of that stride (the beginning of a new stride) is nearly identical to that in the starting conditions. A periodic or cyclic walking motion will then result. The original model for passive dynamics is based on human and animal leg motions. Completely actuated systems, such as the legs of the Honda Asimo robot (Fig.1, right), are not very efficient because each joint has a motor and control assembly. Human-like gaits are far more efficient because movement is sustained by the natural swing of the legs instead of motors placed at each joint. Passive dynamics is a valuable addition to the field of controls because it approaches the control of a system as a combination of mechanical and electrical elements. While control methods have always been based on the mechanical actions of a system, passive dynamics utilizes the discovery of morphological computation [18]. Morphological computation is the ability of the mechanical system to accomplish control functions. The dynamic modeling plays a very important role in such studies and it is being used as a tool when designing structure of the system and its control unit. In this case we talk about simulation. The other application of dynamic molding is in on-line control of the system which is also called dynamic control. While for simulation one should generally use the best available model, the control can be based on a reduced dynamic, depending on a particular task. In this paper, we present the dynamic modeling and Matlab based numerical simulations of a passive robotic walker using Euler-Lagrange method. The main objective is to derive a generalized set of equations of motion and their associated transition equations to study the dynamics and behavior of a simple passive walking. We derived the equation of motion using Euler-Lagrange method which is based on the formulation of system dynamics in the form of kinetic and potential energy stored. We demonstrate that a simple, uncontrolled, 2D, two-link model, roughly resembling human legs, can walk down a shallow slope, powered only by gravity. We also used the same model to simulate its active version using the same dynamic model but powered by an actuator at hip instead of slope. Our model has two rigid massless legs hinged at the hip, a point-mass at the hip, and point-masses at the feet. After nondimensionalizing the governing equations, the model has only one free design parameter, the ramp slope (γ). The rest of the paper is organized as it follows. In Section 2, we describe the model of the biped walker, the derivation of its equations of motions using Euler-Lagrange approach and the transition equations. We also demonstrate the implication of the passivity property to active walker which can walk on level ground without trajectory tracking. Matlab based numerical simulation results are shown in Section 3. Finally, in section 4, the conclusion and future work are outlined.. II. THE BIPED MODEL AND DYNAMIC EQUATIONS A. Model Derivation The main motivation in this paper is to derive a generalized set of equations of motion and their associated transition equations. The sketch of our simple passive walker is shown in Figure 2. It has two rigid legs connected by a frictionless
hinge at the hip. We considered the following assumptions in our model x The legs and the hip are considered point masses x The stance leg is connected to the ground during a step (no slipping) x The robot has point feet with a single contact point with the ground The parameters and variables have been shown in Figure 2. In general, the motion of equation for the biped robot on slope ( ) can be written as ( ) ̈+
, ̇
̇ + ( ; )=
(1)
Where, ( ) is the mass matrix, , ̇ is the Coriolis matrix, ( ; ) is the gravity and is the actuation torque. Note, in case of passive walker, the torque applied by the actuator is zero, thus ( ) ̈+
, ̇
̇ + ( ; )=
(2)
The Euler-Lagrange method is used to find the equation of motion for the biped robot. The Lagrangian formulation describes the behavior of a dynamic system in terms of kinetic and potential energy stored in the system rather than of forces and moment of the individual members involved. The constraint forces involved in the system do not appear in the formulation of Lagrangian dynamic equations. The closedform dynamic equations can be derived systematically in any coordinate system. In general, there are following steps for Lagrange equation of any system x Determine the number of degrees of freedom in the system and select appropriate independent generalized coordinates. x Derive the velocity for the center of mass of each body in term of the generalized coordinate position and velocity. x Identify both conservative and non-conservative forces. x Derive the kinetic and potential energy in term of the generalized coordinate position and velocity. x Substitute these quantities into Lagrange’s equation and solve for the equations of motion. , , …… be generalized coordinates that Let completely locate a dynamic system. Let and be the total kinetic energy and potential energy stored in the dynamic , ̇ can be defined as system. The Lagrangian , ̇ =
, ̇ −
( )
(3)
Using Lagrangian, the equation of motion for the biped can be written as , ̇ , ̇ (4) − = ̇ Deriving the and for the dynamic walker shown in Fig. 2
( ) ( + ) − )
+ (
= +
, ̇ =
1 ( 2
(
(5)
− )
+
(6)
+
In order to simulate the walker’s motion, we need to integrate equations of motion and use transition equations. We have two links, so there will be two equations for angular momentum. First we define the impact surface which can be defined as
Where
From
(7)
) ) − ) )
( ) ( ) ̇
Where
(8)
ℎ ( )=
(− )
(17)
(9)
ℎ ( )=
( )
(18)
ℎ ( )=
( )
(19)
( )=
(− ) +
we can write (10) ̇
̇ =
(16)
( )=ℎ ( )−ℎ ( )+ℎ ( )=0
( =− ̇ ( ( ) = − ( ) acos( =− ̇ asin(
̇ , ̇ =
1 2 ̇
(11) ̇
( )
(20)
Substituting these expressions ( ) ( ) (− ) − = sin( ) ( ) (− ) + =0 + cos( )
(21)
( ) ( ) ( ) − cos( ) = cos( ) ( ) sin( ) = 0 ( ) sin( ) + −
(22)
After simplification ( ) = cos( +
) − cos( +
(23)
)=0
We considered the following assumptions to derive the transition equations
Figure 2 A simple passive walker model ( = = ). The parameters and variable have been shown.
Where, the mass matrix, can be extracted as ( ) + ( + ) − ( − ) = − ( − )
(12)
x
The impact forces the foot of the swing leg experience during the collision with the ground can be represented by impulses.
x
Due to the impulsive nature of the impact forces, the robot configuration remains unchanged during the collision with the ground.
This means that angular momentum around the impact point is conserved Angular momentum of a point mass can be written as =
Angular momentum of the walker about the impacting foot
Introducing KE and PE in the Euler-Lagrange equation we obtain also =
, ̇ =
0 ̇ sin(
− −
)
̇ sin( 0
(1,1) ( + )
( ; )=
−
)
(13)
( (
+ )− + )
+ )
×
=
Where the position vectors by (14)
(
( )
(15)
( )
×
,
(25)
∈ {1, 2, }
( )
(1,1) = −2 +
(24)
×
−asin( ) = acos( ) 0
( )
relative to this point is given
(26)
( )
( )
−lsin( ) = lcos( ) 0 − sin(− ) ( ) = + −bcos(− ) 0
(27) (28)
Where, and are velocities before and after the impact respectively, which can be obtained from followings 0 = 0 × ̇ 0 = 0 × ̇
(− (− 0 (− ( 0
0 = 0 × ̇
(− ) (− ) 0
=
0 − + 0 × − ̇
=
0 + 0 × − ̇
0 − = 0 × ̇
) )
(29)
) )
(30)
(31) (− ) ( ) 0
( ) ( ) 0
( ) ( ) 0
(32)
(33)
(34)
By considering the conservation of angular momentum of the post-impact swing leg around the hip. =
=
− −
× (− ) ( ) 0
=
×
(35) (36)
B. Active Dynamic Walker on Level Ground Based on Passive Property The model presented so far is the model of completely passive walker. The ultimate objective of studying passive dynamic walkers is to develop highly efficient active dynamic walkers. Passivity property can be extended to active dynamic walkers on level ground. One remarkable feature of such active control is that it does not require trajectory tracking. The zero moment point (ZMP), a critical issue in active biped robots that use trajectory tracking controllers, is autonomously regulated by the control law to yield a stable gait. Most of the passive dynamic walkers studied till date have either arc-shaped feet or point feet. Their corresponding active dynamic walkers have arc feet with hip actuation or flat
feet with ankle and hip actuation. We used the former approach to simulate the dynamics in the active version of the walker. Since the gait of passive dynamic walkers requires very low energy inputs through gravity. One of the straightforward ways of achieving level ground walking without destroying the passivity property is to imitate walking on a virtual slope. The virtual gravity field can be defined by an acceleration of / γ , under which the biped robot exhibits passive walking with steady gait. Hence, highly efficient active dynamic walkers can be developed with the associated passivity property by means of minimal control efforts and without disturbing the stability. It is well known that. We used our dynamic model for the numerical simulation of biped walkers as illustrated in Section III. III. NUMERICAL SIMULATIONS To find stable walking gait and test the walking dynamics derived above, numerical simulations were performed using MATLAB. The passive robot can walk stably down a gentle slope by providing a proper initial condition to the robot. The proper initial condition here is the so-called fixed point. If in every step the passive walking model is converged to the same initial condition, this initial condition is called the fixed point. The well-known Newton– Raphson iteration algorithm is used to find the fixed point here. The walking cycle are computed on the basis of the principle of angular momentum conservation. The walking cycle are numerically integrated using ODE 113 method until the running program detects the impact event of the impact phases. The simulation is started when the stride is just beginning and both legs are momentarily on the ground. The back leg is about to take off. The subsequent motion of the legs can now be found by numerically integrating the equations of motion describing the walker. The swing leg will at some point in time hit the ground again; the end of the stride. Assuming that this heel strike is inelastic, the foot will stay on the ground. The impact changes the velocities of each leg. Now, the model is poised to begin another stride. If speeds and angles at this instant are equal (but mirrored) to their values at the beginning of the previous stride, then the model has hit upon a passively re-entrant cycle and can theoretically keep walking indefinitely. If small errors are inserted in this cyclic motion, they could grow with each stride if the motion is unstable or decay, and eventually disappear over a number of strides if the motion is stable. The physical parameters used in simulations are listed in Table 1. Fig. 3 shows the snapshot of the simulation of a simple passive walker while Fig. 4 shows the numerical simulation results of its stance and swing angle. The Fig 5 and 6 shows the pictures of the simulation of active underactuated biped walker. The walker has actuation at hip joint and it uses the arc feet. TABLE 1: PHYSICAL PARAMETERS OF THE WALKER USED IN NUMERICAL SIMULATIONS 10.0 kg 1.0 kg 0.5 m 0.5 m 1.0 ( + )
9.8 0.01 rad
Figure 3 The motion sequence: Snapshot of the simulation of a simple passive walker
Figure 6 Numerical simulation results of stance and swing angle during a powered underactuated walker.
IV. CONCLUSION AND FUTURE WORK
Figure 4 Numerical simulation results of stance and swing angle during passive biped walker.
Figure 5. The snapshot of the simulation of a powered underactuated walker with arc feet.
Humans as bipeds enjoy certain advantages over other terrestrial systems, which motivate us to study and develop biped robots. Underactuated biped robots adopt the energy efficient gait of the biological counterparts and passive walkers. Autonomous walking bipedal machines, possibly useful for rehabilitation and entertainment purposes, need a high energy efficiency, offered by the concept of passive dynamic walking. In this paper, we present numerical simulations of a completely passive walker and its active version by exploiting the passive dynamics. We derived the equation of motion using Euler-Lagrange formulation. We demonstrate that a simple, uncontrolled, 2D, two-link model, vaguely resembling human legs, can walk down a shallow slope, powered only by gravity. Our model has two rigid massless legs hinged at the hip, a point-mass at the hip. This model shows stable walking modes similar to more elaborate models, but allows some use of analytic methods to study its dynamics. We also used the same passive dynamic model to simulate its active version but powered by an actuator at hip and arc feet. Though our model is quite simple but it captures most of the essential benefits of simple passive walker. We believe that it will be a good starting point for analyzing more complex models with leg mass, torso etc. In future, we are aiming to further investigate various designs, models and control strategies used to enable stable walking and running for the underactuated biped robots. We will explore how the mechanism of such bipeds evolved to incorporate the design variations which significantly improved the system performance.
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