^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

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.

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 [

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 [

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 [

The problem discussed here is depicted in Figure

Oscillating constrained bar.

The mass in which the rigid bar is pivoted (

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;

According to [

Equations (

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

As soon as these two variables are known, the remaining variable,

The equations for the impact are formulated for point

Velocities.

Impulses.

For each of the three rigid bodies, we can formulate now the linear impulse/linear momentum equations in the two directions

To better distinguish between velocities right before and right after impact, they are denoted with superscripts “

And, to be more general, it is also allowed initially for the rigid bodies with masses

For the wall, it is obtained that

For the bar, it is obtained (

These equations simplified if the following assumptions are made.

The external two linear impulses

The rotational motion of both, the wall and the lower rigid body, is omitted; therefore, one has

The wall is allowed to move only in the vertical direction, as well as the lower rigid body; therefore,

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,

The contact zone between the free end of the bar and the wall surface is also assumed ideally smooth; therefore,

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

In order to calculate the velocities at the point of impact,

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

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 (

Initially, all the impulses are obtained.

Comparing (

And with

The denominator of these twoequations then is written as

In order to check (

The values for the parameters used in the numerical simulations that follow are presented in Tables

Numerical values considered in the numerical simulations for different values of

Parameter | Value | Unity |
---|---|---|

2.00 | Kg | |

5.00 | Kg | |

10.00 | Kg | |

5.00 | Nm | |

10.00 | ||

400.00 | Nm | |

1000.00 | ||

7.00 | Ns/m | |

1.00 | Ns/m | |

1.00 | m | |

0.60 | m | |

0.50 | m | |

10.00 | Nm | |

0.1667 |
Kg/m^{2} |

Numerical values considered in the numerical simulations for different values of

Parameter | Value | Unity |
---|---|---|

2.00 | Kg | |

1.00 | Kg | |

1.00 | Kg | |

400.00 | Nm | |

5.00 | Nm | |

7.00 | Ns/m | |

7.00 | Ns/m | |

1.00 | m | |

0.60 | m | |

0.50 | m | |

5 | ||

10 | Nm | |

20 | ||

0.1667 |
Kg/m^{2} |

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 ^{o}, 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

When first contact takes place,

According to Figures

Table

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 (

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