A simple mathematical model for a constrained robotic manipulator is investigated. Besides the fact that this model is relatively simple, all the features present in more complex problems are similar to the ones analyzed here. The fully plastic impact is considered in this paper. Expressions for the velocities of the colliding bodies after impact are developed. These expressions are important in the numerical integration of the governing equations of motion when one must exchange the set of unconstrained equations for the set of constrained equation. The theory presented in this work can be applied to problems in which robots have to follow some prescribed patterns or trajectories when in contact with the environment. It can also de applied to problems in which robotic manipulators must handle payloads.

1. Introduction

There are several ways to deal with the problem of interaction between bodies. Impact dynamics and continuous contact between bodies can both be included in the mathematical model of the constrained problem, or just one of these effects can be considered. It depends, obviously, on the characteristics of the studied problem.

The investigations about the contact between bodies include (at least) two different kind of analysis [1]: one associated with the beginning of contact and one associated with its termination. In the first analysis, the distance between the bodies must be checked in order to know when contact occurs; in the second analysis, once the contact is established, the reaction (normal; compression) force between the bodies must be checked. In the second analysis, contact finishes when the contact force is equal to zero.

One of the hardest parts in the study of contact problems involves the different models that must be developed for contact and noncontact situations and the switching between these models when integrating the equations of motion [2, 3]. The unconstrained problem and the constrained problem do not have the same number of degrees of freedom. Dynamic systems when constrained have less degrees of freedom than when unconstrained.

The transition between constrained and unconstrained motions is sometimes called contact (including impact) and sometimes called just impact (mostly when the bodies separate after the collision). When contact occurs, the new velocities of the bodies involved must be known in order to generate the initial conditions to the second part (constrained problem) of the numerical integration. In the constrained problem, the concept of coefficient of restitution is very important [4].

2. Geometric Model of the System and Governing Equations of Motion

The problem discussed here is depicted in Figure 1. According to this figure, in a part of its trajectory, the free end of the bar moves along the constraint represented by the mass named . All the movements occur in the horizontal plane. When contact occurs, impact and bouncing are also allowed to occur.

The mass in which the rigid bar is pivoted () oscillates when excited by the movement of the bar (free and constrained). In the axis , passing through the connection between the bar and (perpendicular to the paper sheet), there is a prescribed moment, , acting to turn the bar.

The dashed lines represent the position of the masses in which the springs and dampers are free of forces. The dotted line represents the position from which one starts to count the angular displacement, .

In physical terms, this system may represent a robot with a translational joint and a rotational joint; can be thought as an obstructing wall on the robot’s trajectory (or some object this robot must handle or interact with), and can be thought as an external torque provided by a dc motor.

According to [5], the constrained governing equations of motion for this system are given by and the constraint condition is given by where represents the bar moment of inertia around its center of mass, represents the mass of the bar, represents the distance from to the cm of the bar, represents the damping coefficient of , represents the damping coefficient associated with mass , represents the stiffness coefficient of mass , represents the stiffness coefficient associated with , and represents the amplitude of the normal force. It is assumed the there are no friction forces involved and represents the total length of the bar.

Equations (2.1) are the equations of motion for , , and . Equation (2.2) is an additional relationship between the generalized coordinates , and when contact occurs. Equations from (2.1) to (2.2) provide four equations and four unknowns (, , , and ) considering the constrained problem and three equations and three unknowns (, , and ) considering the unconstrained problem. In the unconstrained case, (2.2) does not apply and .

3. The Contact Case

In contact, for this problem, there is the loss of one degree of freedom. In other words, one of the variables is dependent on all the others. The best choice is the elimination of the generalized coordinate , which is not always present into the system represented by the oscillating bar [6]. The new set of equations [5] is given by The fully plastic impact case is considered here for the calculation of the velocities immediately after contact. Separation will take place when the normal force is zero.

As soon as these two variables are known, the remaining variable, , is also known through (2.2). Equations (3.1) represent, respectively, the time behavior of the generalized coordinates and during the contact condition. In [5], an analytical expression to the reaction force, , is also presented.

4. The Determination of the Velocities after Contact (Impact)

The equations for the impact are formulated for point (see Figure 2 for the representation of the velocities of the three bodies) where, for sake of clarity, it is distinguished between Point belonging to the wall and point belonging to the bar. Figure 3 shows the free body diagram for the three rigid bodies indicating not the forces at the points of connection or contact but rather indicating the equivalent linear impulses due to impact. All these quantities are marked with an overhead symbol “hat”, for example, , which is the linear impulse of the equivalent force . The physical dimension is the same as the linear momentum, that is, , except for the angular impulse whose unit is .

For each of the three rigid bodies, we can formulate now the linear impulse/linear momentum equations in the two directions and . Additionally, for the rotating bodie(s), we have the equivalent angular impulse/angular momentum equation in -direction, formulated w.r.t. to the respective centre of mass.

To better distinguish between velocities right before and right after impact, they are denoted with superscripts “” (after) and “” (before). Their two components in - and -directions are indicated by corresponding subscripts “” and “”.

And, to be more general, it is also allowed initially for the rigid bodies with masses and to rotate as well. The respective angular velocities therefore will be denoted by with appropriate indices. Later, this additional degree of freedom will be kinematically constrained.

For the wall, it is obtained that

For the bar, it is obtained () that And, finally, for the lower rigid body with mass , it is obtained that assuming that the directions of and are going through the center of mass. The geometric quantities and , not shown in Figure 3, denote the distances of the respective linear impulses measured from the center of mass.

These equations simplified if the following assumptions are made.

(1)The external two linear impulses and , and the angular impulse are small compared with the internal impulses; therefore, they can be neglected.(2)The rotational motion of both, the wall and the lower rigid body, is omitted; therefore, one has and .(3)The wall is allowed to move only in the vertical direction, as well as the lower rigid body; therefore, and .(4)The contact surface between the lower rigid body and the left or right vertical guiding surface (not shown in the figures) is assumed ideally smooth; therefore, . (5)The contact zone between the free end of the bar and the wall surface is also assumed ideally smooth; therefore, . Otherwise, if this surface is rough, we have to account for an additional velocity relationship, for example, given by the definition of the coefficient of restitution in -direction.

Applying these assumptions, the following set of equations is obtained:

In order to calculate the velocities at the point of impact, , only (4.4) and (4.7) to (4.10) are of interest. Additionally, it is needed to establish some kinematic relationships. For the bar center of mass, one has where the length of the vector is just . Equation (4.13) is valid for the velocity right before and after impact. For the free end of the bar, it is obtained equivalently with . In the same way as (4.13), equation (4.14) is valid for the velocity right before impact and right after. During impact, one has the additional equation, which relates the velocities before and after impact at point , in the direction normal to the contact surface, that is, in -direction: with

In the following, it is assumed that there is a fully plastic impact, that is, the impacting bodies maintain steady contact as far as the contact force is repulsive (otherwise, they will separate). This leaves , and hence or

With these equations, it is possible to calculate all the velocities right after impact, given the velocities before impact. Additionally, but not needed here, it is also possible to calculate the appropriate linear impulses. To summarize, one has the following eight equations to determine all the five velocities right after impact (), as well as the impulses ():

Initially, all the impulses are obtained. is simply obtained from (4.23) or by adding the two (4.19) and (4.20), giving also goes simply with (4.21), and is simply obtained directly from (4.19) or by adding (4.20) and (4.23)

Comparing (4.27) with (4.29), it is observed that both equations yield the same result for the linear momenta before and after impact. To determine now the velocities right after impact, one can rely on (4.22), (4.24), (4.25), (4.26), and (4.27) (or (4.29), which is the same). Replacing , and , one arrives at the two equations for the unknown velocities and :

And with , and , these equations can finally be expressed by means of the independent velocities, , , and , right before impact: With the abbreviations one finally obtains

The denominator of these twoequations then is written as

In order to check (4.33), one case is investigated; that is, for , we should maintain the simple translational impact between the combined rigid body consisting of the two masses and and the wall with mass . For the fully plastic impact, one then obtains from (4.33) with and : where the first equation for the translational motion coincides with the result governed from simple impact of two rigid bodies.

5. Numerical Results

The values for the parameters used in the numerical simulations that follow are presented in Tables 1 and 2. The time step considered in the integration of the governing equations of motion is kept constant and equal to 0.0001 s. The fourth-order Runge-Kutta is the numerical integrator used. Two different classes of simulation are investigated.

The constant torque (with different amplitudes) was chosen because it is the simplest one, and in order to make the bar rotate always in the same direction and fulfill . Any other kind of excitation (e.g., like a sinusoidal one with maximum amplitude of 180o, for instance) can be chosen without problem. In the simulation runs, the motion of the bar starts always in its horizontal position to the right, that is, with .

The very beginning of contact is considered here as a fully plastic impact with impact time and with , where represents the coefficient of restitution. Contact finishes when . No friction or contact is considered, up to this point of the investigation, between and the guide it slides through or between and .

5.1. Considering Different Values of

When first contact takes place, is at rest. The second contact (only shown here for the simulations varying ) will happen with presenting some velocity. The bar is able to develop many turns and, in fact, there are possibilities for it to reach many contact conditions as the time evolves.

According to Figures 4, 5, 6, 7, 8, and 9, the amplitude of jumps at the beginning of contact, from zero (no contact) to a value associated with the impact force between the bodies. The contact force evolves with time according to the system states and properties. The value of at the instant of impact does not necessarily represent the biggest value for the contact force, as can be seen in these figures. A sudden change in velocity, when collision takes place, can be verified clearly in these figures.

5.2. Considering Different Values of

Table 2 shows numerical values considered in the numerical simulations for different values of .

6. Conclusions

To conclude, it is important to say that the time step used in the numerical integration and the choice of the integrator are very important aspects to be considered. New numerical integrators can be tested in the course of this investigation and results compared to the ones presented here.

An important consideration not to be forgotten when dealing with problems presenting some sort of constraint is that more than one set of governing equations of motion must be integrated to cover all the system dynamics. The set of equations that governs the system dynamics when the constraint condition is active is different from the one that governs the unconstrained movement of the system. One of these sets is always generating the states for the other.

In this context, the determination of the velocities after contact (impact) is very important. The velocity expressions presented in (4.33) are the necessary corrections one must do when considering the fully plastic impact case. If this correction is not taken into consideration in the numerical integration of the governing equations, the system will gain energy after impact, which is not true.

It is important to realize also that the number of degrees of freedom involved changes from one set of equations to the other. The necessity for changing from one set of governing equations to another (according to the system’s requirements of contact or noncontact conditions) represents a source of integration errors, since the integrator is faced with singularities.

The problem presented in this paper and the procedures developed for its analysis can be extended to many other systems and situations (including more complex ones). The theory presented here can be applied to problems in which robots have to follow some prescribed patterns or trajectories when in contact with the environment (like in painting activities, for instance, or the ROKVISS experiment at DLR).

The next steps are the development of the analytical expressions for the velocities after impact considering any value for the coefficient of restitution and the inclusion of friction forces between and the left and right vertical guiding surfaces; and between the free end of the bar and .