Most mobile robots designed and built use wheels for locomotion. Two of the most important points when designing reliable legged robots are stability and energy efficiency.
Motivation
The ability of an actively balanced system to depart from static equilibrium relaxes the rules on how the legs can be used for support, a fact that significantly improves the robot's mobility. In simpler words, the control action aims to try to help the robot move in the way it wants to move, by exciting its passive dynamics ie.
Literature Review
- History of Legged Machines
- Models for Legged Locomotion
- Dynamic Stability Analysis
- Passive Dynamics
To control the forward speed of the monopod, the control system places the toe in a desired position relative to the center of mass during flight. When walking, the center of mass vaults over a rigid leg, analogous to an inverted pendulum, see fig. 1.3. Like a pendulum, the body's kinetic and gravitational potential energies are cyclically exchanged.
Objective
Passive dynamic boundary walking in quadruped robots was first reported in [8]. Passive dynamic boundary transitions are periodic transitions and can start at stable or unstable fixed points. Stable corridors require no control input and can tolerate disturbances. corridors can be stabilized by the application of appropriate control inputs. Whether a periodic passage is stable or unstable is determined by the eigenvalues of the Poincare map. ) where they exist is limited.Controller for stabilizing gaits from unstable fixed points is an active area of research [11].
Outline of Thesis
Physical Parameters
Physical parameters like body mass, leg mass, spring stiffness, inertia etc. are taken from Scout II and are shown below.
Virtual Leg concept
The axial force exerted by the springs of each of the physical legs must be half the force exerted by the spring of the corresponding virtual leg. The feet of the physical legs that form a virtual leg must touch the ground and leave the ground at the same time. The forward position of the virtual leg's feet relative to the hip must be the same as the forward position of the physical legs' feet.
Bounding Gait
Equations of motion
The direction of this force is along the leg, where Fx and Fy are the components of this force along x and y axes respectively [11].
Touch down and Liftoff Events
Conclusions
Legged robots are hybrid systems with discrete transformations that control transitions from one phase to another phase of movement [8]. Poincare map can be used to determine the stability of the robots as it replaces another order continuous-time autonomous system by an (n- 1)th order discrete-time system. The problem of studying the stability properties of a periodic solution of a continuous-time system is thus reduced to the problem of studying the stability of the periodic points of the Poincare map. To define the return map for a bone system, a reference point in the cyclic motion must be selected and then the dynamic equations must be integrated from that point to the next cycle. Here we take vertex as initial condition i.e. ˙y2= 0 .Since x is monotonically increasing here, it is irrelevant for a periodic trajectory, so we are left with only four variables y, θ,x,˙ θ.˙.
Finding Fixed points
The roots of the above equation that satisfy periodicity are called fixed points. For a given back and front leg contact angles, Newton-Raphson method can be used to find roots of the equation with a proper initial guess. To find a solution iteratively, the convergence value is set to 10−5 and the absolute, relative tolerances are both taken as 10−9. The initial guess for the fixed point is updated using the equation. P = [ δP/δy δP/δθ δP/δx δP/δ˙ θ˙ ] (3.4) Stability of fixed points can be found by examining the eigenvalues of Jacobian matrix of return map P . One of the eigenvalues is always unity, which reveals the conservative nature of the system [2]. Stability of fixed points depends on whether the remaining eigenvalues are inside the unit circle (stable) or outside the unit circle (unstable).
Properties of Fixed points
Control Law
Body Fixed Touchdown angles
Instead of using dip angles measured with respect to absolute vertical, relative dip angles have these advantages. Figure.3.1, shows the stability region with the relative launch angles of the rear leg against the pitch angle velocity at the apex for forward speeds of 1m/s and 2m/s [11].
Conclusions
When both legs are in contact with the ground, the robot body and the two legs form a four-bar mechanism with the ground as a fixed link. The movement of the robot requires both legs to rotate in the same direction around their respective contact points. Double support phase is therefore not allowed when legs are physically cross-connected. The proposed quadruple model (in 3D) with cross-coupling between front and rear legs is shown below.
In this proposed model, considering the foot mass as well, we have three joints i.e. torso, front leg and rear leg which participate in the dynamics at each stage. So, each connection is assigned with three coordinates x, y, θ as shown in Fig.4.3. The body is shown with a blue line, and the upper front and rear legs are shown with red lines, and the green lines show the springs attached to the upper legs. In the case of massless legs, the rear and front legs can be set at the required contact and takeoff angles without much energy consumption. But in this proposed model as the mass of the leg is taken into account, placing the legs at the required angle requires a significant amount of energy, which deviates the model from the state of passive dynamics and makes the energy inefficient. So, only the initial conditions of the peak model are given and then its stability is tested. The model is considered failed if any event does not occur within a reasonable time or the double support phase occurs.
Various Phases in Bounding
Equations of motion
During the stance phase of the hind legs, the spring is compressed and decompressed to its original value, so that the spring force also participates in the dynamics. We also have an additional translational constraint compared to the flight phase. The quadruped model in the stance phase of the hind legs is shown below,. During the front leg stance phase, resilience also plays a role in the dynamics. The quadruped model in the front leg stance phase is shown below.
Phase Transition
Flight phase 2 ends with the landing of the foreleg and the reaching of the stance phase of the foreleg. The stance phase of the foreleg ends with the lifting of the foreleg and flight phase 3 is reached.
Impact Modeling
Using the method of Lagrange, the dynamic model of the robot, [16] can be written as. When the spring comes into contact with the ground, the velocity vector suddenly changes, [14]. Using the conservation of linear and angular momentum and momentum, one can write the impulsive dynamical model as.
Energy loss due to impact
As energy is lost in each cycle, the robot will eventually fail over time. So, to cope with this energy loss, a torque of -0.08 N is provided throughout the stance phase of the back for each cycle, from the actuator to the hip joint between the torso and the back.
Error plot
Conclusions
Fig.5.1 shows the stability region where the robot could bound more than 5 cycles and Fig.5.2 shows the stability region for bounding more than 10 cycles at different pitch speeds and forward speeds when the initial top height is 0.35 m. It is clear from the figures that the cross-coupled leg-mass quadruped robot is only stable at lower forward speeds up to 1.5 m/s, above which the robot completes less than 10 cycles. Figure 5.3 shows the stability region for more than 10 cycles at different pitch speeds and initial hind leg angle (keeping all other initial conditions unchanged) when the initial top height is 0.35 m and 0.4 m.
From the Fig.5.4 we observe that by changing only the initial leg angle while keeping all other initial conditions unchanged, the stability region varies which clearly depicts the sensitivity of the robot to the initial conditions. From the Fig.5.3 and Fig. 5.4, it is clear that the stability region is more for top height of 0.35 m than top height of 0.4 m.
Application of torque to increase stablity
Using GUI
The value property contains the numeric value of the slider. The Max and Min properties specify the range of the slider. The slider property controls the amount of change in the slider value when the slider is moved. Using the editText component, the user can set the value by typing the required value in the edit box which will send the sliders callback to get the value set by the user. If the user specifies a string or a value that is not in the slider's range, then the slider defaults to the slider's minimum value. Figure.5.7, shows the GUI application, where two sliding components with a range from 0 to 1 Nm are used to adjust the torque applied in the stance phases. The stability region of the hybrid quadruped robot bounded by leg mass is smaller when compared to the stability region with massless legs.
The idea of cross-coupling between legs is only useful at forward speeds of less than 1.5 m/sec. The cross-coupled quadruped model can operate more cycles by continuously adjusting the torque applied in each stage using human intelligence.
Future Work
Poulakakis Iaonnis, "On the Passive Dynamics of Quadrupedal Running", Master's Thesis, McGill University, Montreal, Canada, 2002. On the stability of the passive dynamics of quadrupedal running with a limiting gait.” The International Journal of Robotics Research (2006). H., “Hopping in Legged Systems – Modeling and Simulation for the Two-Dimensional One-Legged Case”, in IEEE Tr.
Sivanand, “Passive dynamic quadruped robot constrained by front-back leg connection”, MSc thesis, Indian Institute of Technology Hyderabad, India 2014.
