On the Dynamic Stability of Mobile Manipulators

Dynamic stability is an important issue for vehicles which move heavy loads, turn at speed, or operate on sloped terrain. In many cases, vehicles face more than one of these challenges simultaneously. This paper presents a methodology for deriving and reacting to proximity to tipover for autononmous field robots which must be productive, effective, and self reliant under such challenging circumstances. The technique is based on explicit modelling of mass articulations and determining the motion of the center of gravity, as well as the attitude, in an optimal estimation framework. Inertial sensing, articulation sensing, and terrain relative motion sensing are employed. The implementation of the approach on a commercial industrial lift truck is presented. 1 Notation 2 Introduction This paper addresses the issue of detecting proximity to a tipover event for a mobile robot operating in a dynamic environment. Mobile manipulators can alter their configuration in response to commands issued to move the payload. As a result, the location of the center of mass (cg) of the vehicle is dynamically located; a characteristic that may adversely affect the mobility of the vehicle. In this work, the general idea of measuring the instantaneous motion of the cg of the vehicle relative to an inertial reference frame is addressed first. Following this, an algorithm to quantify the stability of the vehicle based on the inertial forces acting on the vehicle as a result of induced accelerations is presented. This algorithm is based on the assumption that the internal D’Alembert forces can be neglected. 2.1 Related work Existing approaches belong to the class of quasi-static stability. In them it is assumed that the vehicle motion relative to an inertial reference frame is slow, and that no inertial accelerations are present except gravitational acceleration. As a result, the vehicle does not experiences inertial forces that might contribute to cause the tipover event. In these approaches, the stability of the vehicle is a function of the location of the vehicle’s cg relative to the geometric footprint (called polygon of support) described by the vehicle’s contact points. As the vehicle articulates, the cg of the vehicle moves within the limits of its polygon of support. The net resultant force acting on the cg (e.g., weight) is compared against normal vectors defined for the support polygon. The normal vector that has the largest component along the resultant force will result in the direction of marginal stability. Morover, when measuring the direction of the gravity vector relative to the vehicle it is assumed that the accelerometer indications are those of the vehicle cg. For vehicles that can articulate significant mass over long distances, this assumption becomes invalid since the motion of the vehicle cg can be large. This poses two problems: 1) the cg of the vehicle experiences inertial accelerations and 2) the specific force reported by the accelerometer will not include such accelerations. Therefore, even for a quasi-static stability assumption, if the vehicle can articulate significant mass current approaches are not readily applicable. 2.2 Approach This paper improves on the existing approaches by relaxing these assumptions. In particular, we allow for inertial accelerations that generate inertial forces large enough to cause the vehicle to tipover: dynamic stability. Furthermore, in our approach we measure the instantanous motion of the CG and compute the inertial forces acting on the cg as a result of this motion. To this end, in our approach we make no assumption on the sensor indications. Sensor indications are mapped back to the instantaneous location of the cg, providing true cg motion. 1. Corresponding author. To evaluate the dynamic stability of the vehicle, we define two different vehicle states: motion state and configuration state. The former provides information as to what the vehicle is doing (i.e., how fast it is turning, is it driving on a non-level road, how heavy is the load, etc.) and the latter provides information as to what configuration the different articulated parts of the vehicle are at any point in time. The configuration state of the vehicle determines the limits on the D’Alembert forces that can be applied to the vehicle without actually causing a tipover event. It is also used to compute the instantaneous location of the vehicle cg relative to its polygon of support. To this end, the vehicle and all its articulations and payload are considered to be a lumped body with mass equal to the total mass of the system and with a cg that is dynamically positioned as the vehicle articulates. The location of the cg along with the inertial forces acting on the vehicle enable the computation of the tipover stability margin, which is a measure of the proximity of the vehicle to a tipover event as a function of motion and configuration states. The proposed stability algorithm consists of two parts: 1) an estimation and prediction system and 2) a dynamic module. The estimation and prediction system is aimed at determining the current state of the vehicle using sensors to measure relevant dynamic quantities. The dynamic module is aimed at computing the stability margin using the vehicle’s state estimates. The estimation and prediction module produce an estimate of the motion of the vehicle cg as the vehicle articulates. Since the sensors cannot be mounted near or on the cg and since the motion of the cg can be large (such as when articulating significant payloads), the estimation and prediction system maps the sensor inputs (e.g., speed encoder and accelerometers) onto to the instantaneous location of the cg through successive applications of the Coriolis theorem. The instantaneous location of the cg is computed (using the configuration sensor inputs) by iteratively solving the forward kinematics problem for an n-link manipulator. Once a state estimate is generated, the dynamic module computes the stability margin for that configuration. This process is based on first finding the support polygon of the robot defined by the contact points between support points (e.g., wheels) and the terrain. The edges of this support polygon represent the tipover axes: lines about which the vehicle can sustain an induced rotary motion. This is shown in Figure 1 (that shows a projection of the cg onto the roll plane): the tipover axes are pointing into the page and the vehicle can sustain a rotary motion about each axis. The state of the vehicle together with the instantaneous location of the cg, enable for the computation of a stability margin for each of the tipover axis. The stability margin for the i-th tipover axis is defined as the subtended angle between and . Where is the projection of the net resultant force acting on the cg of the vehicle onto the plane defined by the i-th tipover axis and is the normal of the ith tipover axis that passes through the cg. Finally, the stability margin for the vehicle is defined as . The utility of a tipover proximity indicator is that it can be used to drive a number of mechanicms which can take corrective action. For man-driven vehicles, a console indication or audible warning could be produced. For autonomous systems, an exception can be raised for resolution at higher levels of the autonomous hierarchy or various governing mechanisms can be engaged to actively reduce the severity of the situation. This paper is organized as follows: Section 3 presents the definitions and assumptions used through out the paper, Section 4 presents the system dynamics model used in the optimal estimation framework, Section 5 presents the measurement models that map vehicle motion states onto sensor indications, Section 6 presents our approach to assesing the dynamic stability of the vehicle and Section 7 presents some results from the applicaiton of the algorithms presented in the paper. θi fi li fi li θ min θi ( ) = fr θ1 θ2 l1 l2 Figure 1: Determination of the stability margin of the vehicle. As the vehicle aticulates, and change as a result of the instantaneous motion of the cg. This changes the dynamic stability of the vehicle as the same resultant force is being applied to the cg. θ1 θ2 fr θ1 θ2 l1 l2