Mobile manipulators have reduced maneuverability and risk rolling over when operated at high speeds. One of the main contributing factors is the higher center of gravity (CG) due to the manipulator arm. This paper proposes a new dynamic weight-shifting method that uses the manipulator arm on the mobile robot to improve maneuverability and reduce rollover risk. A control law is developed such that the manipulator arm keeps a low CG and the contribution of the reaction moments from its inertia is small in comparison to the reaction moments due to gravity. A linear dynamic model is used to analyze the effect of the arm design (link length, mass, etc.) on the roll dynamics. A higher fidelity nonlinear simulation is used to evaluate roll reduction and the impact on handling dynamics. Last, the dynamic weight-shifting method is implemented in hardware. With regard to reducing rollover risk, simulation results from the nonlinear model (NLM) show a 29% reduction in wheel normal load transfer by using the proposed method. In terms of improving maneuverability, experimental results with hardware demonstrate a 13% increase in lateral acceleration when using dynamic weight-shifting. By reducing the vehicle's roll motion, dynamic weight-shifting can increase safe operating speeds and maneuverability.

Introduction

Mobile manipulators are used in many fields where it might be impossible or undesirable to put humans. For example, the U.S. Army uses unmanned ground vehicles (UGVs) with attached manipulator arms to dispose of improvised explosive devices or to scout in urban environments ahead of troops on foot. These UGVs enable humans to maintain a safer distance from danger.

However, one limitation with current mobile manipulators is low operation speed. According to Yamauchi, small military UGVs only operate at speeds of up to around 2.7 m/s (6 mph) [1]. It is difficult for human operators to safely maneuver UGVs at higher speeds in comparison to manned vehicles. Information exchanged between the human operator and remote vehicle is often sent over wireless networks that introduce communication delay and limits on bandwidth. This gives the human operator degraded information about the vehicle's environment (e.g., a camera view of the environment used to control the vehicle may have limited field of view, low resolution, and low frame rate). Adding automation onboard the UGV is one way to assist human operators and increase speed. For example, the automotive industry has used differential braking [24], active suspensions [5], and active steering [4,6] to maintain stable vehicle operation at high speeds. However, these methods require decreasing vehicle speed and/or increasing turn radius.

By actively moving the CG of a vehicle, it is possible to improve maneuverability without decreasing speed or increasing turn radius. For example, Yamauchi developed a dynamic weight-shifting system on a high-speed small UGV to correct oversteer and understeer. However, Yamauchi's system required adding additional actuators and hardware to the vehicle to move the CG forward and backward during operation [1]. This raises the question: can one leverage existing hardware on a UGV to safely improve operation speed and maneuverability?

Typically, manipulator arms are kept in static positions while UGVs drive. Previous works suggest using reaction torques due to a manipulator arm's movement to stabilize roll motion (discussed more in Sec. 2). This strategy is useful in high lateral acceleration turns for very short periods of time. However, for long duration turns at high lateral acceleration, the actuators will not have enough torque or large enough stroke to stabilize roll motion. We propose a new dynamic weight-shifting control method for an existing manipulator arm that aids with long duration, high lateral acceleration turns. The new method keeps the manipulator arm's CG low. The reaction moments from its inertia are small in comparison to the reaction moments due to gravity.

Our previous work presented the dynamic weight-shifting method and investigated on how dynamic weight-shifting reduced roll motion in simulation [7]. This paper significantly expands the previous work in two ways. First, it describes how the handling dynamics are affected by dynamic weight-shifting and demonstrates the improvements in maneuverability. Second, the analyses on roll reduction and improved maneuverability are validated in hardware with the experimental platform in Fig. 1.

The remainder of the papers is organized as follows: Section 2 discusses related work. Section 3 describes the dynamic weight-shifting method and how it is modeled in an linear model (LM) and NLM. Section 4 compares simulation results with the LM and NLM. Section 5 presents a sensitivity analysis of manipulator arm parameters, describes how dynamic weight-shifting reduces vehicle roll motion, and improves maneuverability with both nonlinear simulation and hardware experiments. Lastly, Sec. 6 concludes the paper and identifies areas of future work.

Background

Several groups have investigated dynamically moving a manipulator arm onboard a mobile base to increase rollover stability. To accomplish this, they have developed strategies to control the position of the zero moment point (ZMP). The ZMP is “defined at the point on the ground about which the sum of all the moments of active force is equal to zero” [8]. As long as the ZMP is inside the polygon formed by the mobile base's contact points with the ground, the mobile manipulator is stable. Huang et al. initially developed a motion planner for the manipulator using a potential field that drives the ZMP to the center of the stable region as the mobile base drives around [8]. In later work, they developed an improved motion planner that, in addition to maintaining stability, aimed to maintain high manipulability and minimize the manipulator's path acceleration [9].

Kim and Chung investigated a dynamic weight-shifting system that combines the mobile base and manipulator arm subsystems allowing them to maintain rollover stability for both mobile base locomotion and manipulator-oriented tasks [10]. Lee et al. use invariance control and recursive analytic gradients with the ZMP to increase robustness and computation speed in their dynamic weight-shifting control law [11]. These prior works [811] consider simple mobile bases (without suspension) that are limited to relatively low operation speeds (simulation results were at speeds of approximately 2 m/s). In this paper, we consider a high-speed mobile base with a suspension and Ackermann steering (results in simulation up to 15 m/s and in hardware up to 5 m/s).

Patel and Braae have considered rollover prevention of a high-speed mobile base with Ackermann steering. They proposed to add a “tail” to the mobile base to increase maneuverability [12,13]. The tail provides a reaction moment due to a change in angular momentum as it moves to help stabilize the vehicle similar to the function of the arm in Refs. [811]. However, the stabilizing reaction moments can only be applied for short durations due to stroke limit. With our method, the control strategy for the manipulator arm is to keep the CG low and provide a stabilizing moment due to gravity's effect on the arm.

A rollover model is required to conduct our analysis. Modeling rollover of Ackermann steer vehicles is well researched. Models consider two types of rollover: tripped and untripped. According to the National Highway Traffic Safety Administration, in tripped rollover, the vehicle's tires dig into soft soil or strike an object that causes the vehicle to overturn. In untripped, the vehicle does not strike any objects; rollover is induced by a severe maneuver [14].

Untripped rollover models range from very simple one degree-of-freedom (DOF) to very complex models with 14DOF or more [1519]. Many simple rollover models decouple handling and roll dynamics. Rajamani derived a simple 1DOF roll model that includes effects of a roll center offset from the vehicle CG [18, Chap. 15]. This roll model can be coupled with the 2DOF handling model also derived by Rajamani [18, Chap. 2]. Cameron and Brennan describe how a 3DOF model (2DOF handling model and 1DOF roll model) can give a good prediction of performance for an actual vehicle [15]. Chen and Peng discuss the accuracy of several simple rollover models including a decoupled 2DOF roll model with sprung and unsprung masses [20].

A higher fidelity 14DOF vehicle model developed by Shim and Ghike was shown to give very similar outputs to commercial vehicle simulation packages [19]. Many researchers have used commercial vehicle simulation packages such as CarSim or TruckSim to test controllers for preventing rollover [21,22]. However, adding in custom dynamics (e.g., effects of a manipulator arm) can be challenging.

General multibody dynamics simulation packages such as Adams have also been used to develop rollover prevention methods [17]. Chiu developed a vehicle rollover model in simulink SimMechanics [16] and evaluated a differential braking controller to prevent rollover [2]. SimMechanics is also an effective tool for modeling manipulator arm dynamics [23]. The model selected for analysis of our dynamic weight-shifting method will be discussed in Sec. 3.

To understand the effect of our dynamic weight-shifting method, we will compare rollover stability metrics used in the literature both with and without our weight-shifting method. One common measure of rollover stability is related to wheel load transfer. Wheel load transfer metrics, such as those developed by Odenthal et al. [4], are simple and intuitive. Other rollover stability metrics include the force-angle stability measure for low-speed mobile manipulators [24]. Peters and Iagnemma defined a stability moment measure for mobile robots operating at high speeds [25]. Moosavian and Alipour developed a moment-height stability measure [26]. Chen and Peng developed a time-to-rollover metric [20]. Lastly, energy-based rollover stability metrics have been developed based on vehicle roll kinetic energy and tipover potential energy [27]. We will primarily consider wheel load transfer metrics when discussing rollover stability.

Vehicle and Manipulator Model Description

In this work, we consider a high-speed UGV with a manipulator arm attached to it. This type of UGV could be used for a variety of tasks including scouting or retrieving objects. Typically, manipulator arms are kept in a static position during driving tasks, so the dynamic models for driving and manipulation can be developed independently. However, the dynamic weight-shifting method in this paper will need to capture the interaction of moving the manipulator arm while driving.

The UGV base will be modeled as a vehicle with front-wheel Ackermann steering and rear-wheel drive—a typical setup for high-speed UGVs. The manipulator arm will be modeled as a two-link arm with revolute joints and an end effector—an arm representative of those used to retrieve objects.

Two different models will be compared and used to carry out the analysis. The first is a simple 3DOF linearized model (referred to as the LM) that decouples the handling (2DOF) and roll dynamics (1DOF). The LM is used to gain physical intuition of the effects of the manipulator arm and could be used in model-based control methods.

The second is an 11DOF NLM developed in SimMechanics from a vehicle model by Chiu [16]. Chiu's model was chosen because of its high fidelity and flexibility to add manipulator arm dynamics.

Assumptions and Definition for Rollover.

In this paper, terrain roughness is neglected and it is assumed that the mobile manipulator is operating on a flat smooth surface similar to that of a paved road. The inputs given to each model are a steering angle δ and the forward velocity of the vehicle Vx (assumed to be kept constant in the LM). It is assumed that the vehicle roll angle ϕ can be estimated onboard (this estimate of ϕ will be used to control manipulator arm joint angles).

Additionally, since terrain roughness is ignored, we only consider untripped rollover and define the critical rollover condition to be when one of the wheels lifts off the ground (when the normal force Fz on one of the tires becomes zero). The models used are only valid when all wheels are on the ground.

Linear Model (LM).

An LM was first developed to gain physical intuition of the effects of automatically moving a manipulator arm on a vehicle during turning. A simple 2DOF handling model was chosen to determine the lateral acceleration output for a given steering input. The vehicle handling model assumes a constant forward velocity Vx and linear tire model. The input is front wheel steering angle δ and output is lateral acceleration ay.

The lateral acceleration calculated from the handling model is input into a simple 1DOF model used to describe the roll dynamics [20]. The manipulator arm is treated as an element that provides reaction forces and a moment to the vehicle at its point of contact with the vehicle. The 1DOF roll model has an input of lateral acceleration ay and an output of roll angle ϕ. Both the handling and roll dynamics will be presented in state-space form x˙=Ax+Bu, y = Cx with state x, input u, output y, state matrix A, input matrix B, and output matrix C defined in Secs. 3.2.1 and 3.2.2.

Linear Handling Model.

Figure 2 shows the free body diagram for the handling model. The vehicle has yaw inertia Iz, mass m with the CG located a distance f from the front wheel and r from the rear wheel. Thus, the wheelbase is  = f + r. Energy stored in this system is in the form of kinetic energy, so states of lateral velocity Vy and yaw rate ψ˙ are the only two needed, xhand=[Vy   ψ˙]T. The input to the system is the front steering angle δ. The dynamic equations of motion for this planar model result from summing up the moments about the body CG and adding up the forces in the lateral direction of the vehicle fixed frame

 
m(V˙y+ψ˙Vx)=Fyfcosδ+FyrIzψ¨=fFyfcosδrFyr
(1)
In the LM, forward velocity Vx is assumed to be constant. A linear tire model is used such that the front and rear lateral forces can be described as Fyf=Cαfαf and Fyr=Cαrαr, respectively. Note that Cαf and Cαr are the front and rear cornering stiffness. These values are difficult to measure, so they will be tuned to match up the LM handling behavior with the NLM in Sec. 4. The values of αf and αr are the front and rear tire slip angles defined as 
αf=δtan1Vy+fψ˙Vx,   αr=tan1Vyrψ˙Vx
(2)

After substituting Eq. (2) into Eq. (1) and applying small angle assumptions, the result is the state matrix Ahand, input matrix Bhand, and output matrix Chand [18, Sec. 2.3]. Additional details of the parameters used are described in Table 1 of the Appendix.

 
Ahand=[Cαf+CαrmVxrCαrfCαfmVxVxrCαrfCαfIzVxf2Cαf+r2CαrIzVx]Bhand=[CαfmfCαfIz]Chand=[Cαf+CαrmVxrCαrfCαfmVx]
(3)
The transfer function relating the steering angle input to lateral acceleration output can be found from Ghand(s)=Ay(s)/Δ(s)=Chand(sIAhand)1Bhand. The steady-state gain is found by evaluating Ghand,SS=Ghand(iω)|ω=0 and is 
Ghand,SS=CαfCαrVx2(rCαrfCαf)mVx2+CαfCαr2
(4)

Linear Roll Model.

The linear roll model was derived based on Fig. 3. All arrows specifying forces and moments are drawn in the positive direction. Figure 3 is drawn for a vehicle performing a left turn with positive roll angle and positive lateral acceleration. The physical parameters are described in Table 1 of the Appendix. The vehicle is assumed to have a width of 2 T and make contact with the ground on its left and right side. Each side of the vehicle roll model has a stiffness kt and damping bt that contribute to the vertical loads.

A two-link manipulator arm connected to the vehicle at its CG is considered. The arm is assumed to consist of massless links with length L and an end effector with mass mee. Thus, the arm has weight Wee=meeg and increases the total weight of the vehicle to W=(mv+mee)g. With the arm connected to the vehicle, it will provide a reaction moment and reaction forces in the vertical and horizontal directions. The manipulator arm joints will be controlled such that the end effector mass is kept at the same height as the vehicle CG h. Controlling the end effector position in this way will cause the moment arm in the z0 direction to be zero, thus the lateral force Weeay will not contribute to Ma.

In order to keep the end effector position at the same height as the vehicle CG, the joint angles of the arm are selected to be in positions proportional to that of the roll angle ϕ of the vehicle. In this case, joint 1 attached to the vehicle is proportional to ϕ by constant Kϕ, where joint angle Kϕϕ is measured with respect to the z-axis of the vehicle fixed frame. Joint 2 is attached to the end of link 1 and is positioned at an angle of 2ϕ(Kϕ1)180deg measured with respect to the z-axis of the link 1 fixed frame.

With the manipulator arm end effector controlled in this way, the reaction moment due to the arm will be the weight of the end effector multiplied by its distance from the vehicle CG: Ma=2WeeLsin(ϕ(Kϕ1)). Note the following about this control law:

  • Since Kϕ appears inside the sin term and small angle approximations will be used in the LM, the roll model will be more accurate for small Kϕ and small roll angles ϕ.

  • For Kϕ=1, the arm mass will always be located at the CG of the vehicle and contribute no moment. This will be referred to as the “arm stationary” case (Arm Stat.).

  • For Kϕ>1, the arm mass will apply a stabilizing moment to the vehicle as it rolls. This will be referred to as the “arm moving” case (Arm Mov.).

  • For Kϕ<1, the arm mass will apply a destabilizing moment to the vehicle as it rolls. Therefore, we only consider Kϕ1.

The energy stored in this system is in the form of kinetic and potential energy, so the states will be roll angle ϕ and roll rate ϕ˙,xroll=[ϕ   ϕ˙]T. The input to the system is the lateral acceleration ay. The dynamic equations of motion for this planar model result from summing up the moments about the body CG and substitution in the expression for Ma 
Ixϕ¨=2ktT2ϕ2LWeesin(ϕ(Kϕ1))2btT2ϕ˙+Whay
(5)
After applying small angle approximations to Eq. (5), the result is state matrix Aroll, input matrix Broll, and output matrix Croll 
Aroll=[012ktT22LWee(Kϕ1)Ix2btT2Ix]Broll=[0WhIx]Croll=[10]
(6)
The transfer function relating the lateral acceleration input to roll angle output can be found from Groll(s)=Φ(s)/Ay(s)=Croll(sIAroll)1Broll. The steady-state gain is found by evaluating Groll,SS=Groll(iω)|ω=0 and is displayed in Eq. (7). If the arm mass is considered to be stationary at the vehicle CG, then the arm will provide no moment Ma to the vehicle. This case (Arm Stat.) can be treated as Kϕ=1 causing the 2LWee(Kϕ1) term in the denominator of Eq. (7) to become zero. The case where the arm is actively moving (Arm Mov.) to provide a stabilizing moment (e.g., Kϕ>1) will cause the 2LWee(Kϕ1) term in the denominator to be positive 
Groll,SS=Wh2ktT2+2LWee(Kϕ1)
(7)

Nonlinear Model (NLM)

Vehicle Model.

The multibody vehicle rollover model used in this simulation was developed and verified by Chiu in simulink SimMechanics [16]. The model will be briefly described here starting with the main vehicle body shown in Fig. 4. Attached to the vehicle body CG is the standard ISO 8855 coordinate system with the x-axis forward in the longitudinal direction, the y-axis to the left in the lateral direction and the z-axis upward in the vertical direction. Connected to the vehicle body are the front and rear axles. The front and rear axle bodies are connected by 1DOF revolute joints each having their own roll stiffness and damping. Connected to each the front and rear axles are two wheel bodies. The two front wheels are each given a yaw DOF in the z-axis direction to accept a steering input angle δ. Each wheel body has a vertical stiffness and damping and interacts with the road through a 6DOF joint.

The longitudinal and lateral tire-ground contact patch forces are calculated using Pacejka's magic formula tire model [29, Chap. 4]. Parameters for the tire model were obtained from Table 4.1 of Ref. [28]. The magic formula used takes into account the effects of the tire camber angle γ when the vehicle begins to roll. The normal (vertical) force on each tire is calculated at each step of the simulation based on its vertical stiffness and damping interaction with the ground. If the position of the bottom of the tire is calculated to be above the ground, this corresponds to the tire lifting off. Traction force is applied to the rear wheels and the longitudinal speed of the vehicle is controlled using a proportional-integral-derivative (PID) controller.

This model was originally developed with parameters for a full-size commercial van (approximately 2800 kg) in Ref. [16]. In order to make this simulation more relevant to smaller UGVs, the vehicle parameters were selected to be similar to that of the scale RC car in Fig. 1. The RC car weighs approximately 3 kg and is 1:10 scale. Measurements of the RC car that could be easily obtained with standard laboratory equipment were used for the NLM parameters (e.g., wheelbase, track width, vehicle mass, etc.). Parameters that were difficult to accurately measure were approximated by scaling the commercial van parameters. Based on the mass (1:1000) and length (1:10) scales, the commercial van mass parameters were multiplied by 10−3, length parameters by 10−1, inertia values by 10−5 (mass × length2), stiffness and damping values by 10−4 (mass × length), etc. Specific values for NLM parameters can be found in Table 1 of the Appendix.

Manipulator Arm Model.

A simple two-link arm will be simulated to represent a manipulator arm on a UGV. Both DOFs are revolute joints aligned with the vehicle's roll axis. The arm is representative of a simple manipulator that could be used for picking objects up around the UGV's base. The manipulator arm parameters were selected to match the setup in Fig. 1.

The manipulator arm was modeled in simulink using SimMechanics and SimElectronics. The arm consists of two links, each connected by a revolute joint, and an end effector mass. The arm links are approximated as rods with mass mL and inertia (mLL2/12)·diag([1 1 0]). The end effector mass is approximated as a cuboid with mass mee and inertia (meedee2/6)·diag([1 1 1]). Each joint has 1DOF in the x-axis direction of the vehicle-body-fixed ISO coordinate system. Values for the manipulator arm parameters are listed in Table 1 of the Appendix.

The joints are actuated using closed-loop PID position control. Included in the control loop is a direct current motor model from SimElectronics with torque saturation τm representative of an appropriately sized motor. Each motor tracks a joint position based on Kϕ and the current vehicle body roll angle ϕ. The motor on the base joint of link 1 is attached to the vehicle body CG. Thus, reaction forces and torques due to the arm's dynamics are transferred back to the vehicle body.

Model Comparison

The LM and NLM were used to analyze the effects of using a manipulator arm for dynamic weight-shifting during turning maneuvers. The maneuver selected for this analysis was a steering steplike input of magnitude δ with the vehicle moving at constant forward velocity Vx. The steering input profile used for nonlinear simulations in this paper has a value of 0 for 1 s, then increases linearly to the final steering angle δ over a duration of 0.2 s and remains constant.

A comparison of the LM and NLM was performed in steady state at various steering inputs and forward velocities. Steering inputs of δ = 1 deg → 5 deg in 1 deg increments were given to the NLM at forward speeds Vx = 2, 4, 8 m/s. First, the steady-state behavior of the handling models was compared, i.e., the resulting steady-state lateral acceleration for a constant steering input. The result is displayed in Figs. 5(a) and 5(b). As expected, the LM yields a linear relationship of ay versus δ with slope equal to Ghand,SS, which increases with increasing Vx.

The LM and NLM steady-state ay values line up well at Vx = 4 m/s. At Vx = 2 m/s the LM predicts slightly higher steady-state ay values. As forward speed increases to Vx = 8 m/s, the NLM becomes more nonlinear for steady-state ay values versus δ. Since the handling and roll models are decoupled in the LM, the steady-state ay values are unaffected by whether Arm Stat. or Arm Mov. The NLM shows that steady-state ay values are higher for Arm Mov. versus Arm Stat., especially at higher Vx.

Next, the steady-state behavior of the roll models was compared. The result is displayed in Figs. 5(c) and 5(d). Again, the LM and NLM steady-state ϕ values line up well at Vx = 4 m/s. At Vx = 2 m/s the LM predicts slightly higher steady-state ϕ values than the NLM. As forward speed increases to Vx = 8 m/s, the NLM deviates more from the LM steady-state ϕ values.

The transient behavior of the LM and NLM was compared for a forward speed of Vx = 8 m/s and steering input of δ = 3 deg in Fig. 6. Subplots Figs. 6(a) and 6(b) compare the lateral acceleration responses for Arm Stat. and Arm Mov., respectively. The transient responses line up well for Arm Stat., as one could have predicted from Fig. 5(a) (since the NLM “+” for Vx=8 m/s,δ=3deg was close to the LM trendline). For Arm Mov., the LM outputs a lower ay, as one could have predicted from Fig. 5(b).

Figures 6(c) and 6(d) compare the roll angle responses for Arm Stat. and Arm Mov., respectively. The solid lines represent simulation results entirely using the NLM. The dashed lines represent simulation results using the linear handling and roll models. The dashed–dotted lines input the values from the NLM ay transient into the linear roll model. When an accurate estimate of ay is known, the linear roll model lines up closely with the NLM, i.e., the solid and dashed–dotted lines are close. However, when estimates of ay are bad (e.g., dashed line in Fig. 6(b)), the linear roll model will not predict transient behavior well (dashed line does not line up with solid line in Fig. 6(d)).

While the accuracy of the full linear (handling and roll) model suffers at higher forward speeds (Vx8 m/s), it does capture a similar trend to that of the NLM. The LM can serve as a starting point for selecting parameters that will improve roll stability. For instance, the LM can help predict how adjusting the manipulator arm link lengths L, arm end effector mass mee, or control gain Kϕ can improve roll stability (see Sec. 5.1). Since the full LM is not well suited for predicting transient behavior at higher forward speeds, the linear roll model could be combined with a more accurate estimate of lateral acceleration. For example, the lateral acceleration may be able to be estimated from a desired vehicle trajectory. Then, the resulting lateral acceleration estimate and linear roll model could be used in a model-based control method. Future work could also consider a higher DOF LM that captures the dynamic interaction between roll behavior and handling.

Results

Manipulator Arm Parameter Sensitivity Analysis.

As discussed in Sec. 2, many rollover stability metrics exist. Since the critical rollover condition was defined to occur when a wheel lifts off the ground, the brief survey of rollover stability metrics indicates that the normal load transfer metric R defined in Ref. [4] is well suited for this analysis because of its simplicity and relation to the critical rollover condition. This load transfer metric is shown in the following equation, where FL refers to front left and FR to front right tire when sitting facing forward in the vehicle: 
R=|Fz,FLFz,FRFz,FL+Fz,FR|
(8)
The load transfer metric R can be used to calculate the roll reduction factor ρ as defined in Eq. (9), where Rmov. is load transfer metric for the Arm Mov. case and Rstat. is the load transfer metric for the Arm Stat. case 
ρ=Rstat.Rmov.Rstat.
(9)
In steady-state with the LM, the relationship between R and ϕ is found by subtracting the right from the left normal force in Fig. 3 and dividing by the total weight of the system. Additionally, in steady-state the roll rate ϕ˙ is zero. The result is Eq. (10), where ϕss is the steady-state roll angle 
Rlin=|W/2+ktTϕssW/2+ktTϕssW|=2ktTW|ϕss|
(10)
Thus, the load transfer metric is proportional to ϕ by a constant that is independent of whether Arm Stat. or Arm Mov. in the LM case. Therefore, the roll reduction factor can be expressed as a function of only steady-state roll angles for the LM case as shown in the following equation: 
ρlin=|ϕss,stat.ϕss,mov.ϕss,stat.|
(11)
Since the handling and roll dynamics of the LM are decoupled, the steady-state gain contribution from the handling model in ρlin will cancel out. Thus, ρlin is independent of Vx and δ and can be expressed as a function of only the linear roll model parameters 
ρlin=1ktT2ktT2+LWee(Kϕ1)
(12)

For very small L and/or Wee, ρlin will approach zero. For very large L and/or Wee, ρlin will approach one.

With the roll reduction factor developed above, an arm parameter sensitivity analysis was performed. Figure 7(a) displays an analysis of the sensitivity of the manipulator arm link length L. Along the horizontal axis, L is varied between 20% and 300% of its nominal value of L = 0.5 m. The vertical axis displays the normalized roll reduction factor ρ¯, which is defined as the calculated roll reduction factor divided by the roll reduction factor at the nominal case of L = 0.5 m

 
ρ¯=ρρnom
(13)

Figure 7(b) displays an analysis of the sensitivity of the manipulator arm end effector mass mee. Along the horizontal axis, mee is also varied between 20% and 300% of its nominal value of mee = 0.5 kg and the vertical axis displays the normalized roll reduction factor ρ¯ shown in Eq. (13).

Lastly, Fig. 7(c) displays an analysis of the sensitivity of the manipulator arm joint angle constant Kϕ, which is varied between Kϕ=1 (Arm Stat.) to Kϕ=5. The nominal value used in calculating ρ¯ is Kϕ=2.

The NLM was run at test cases of Vx = 4 and 8 m/s with δ = 4 deg. Figure 7 shows that the LM predicts a similar trend as the NLM in the area around the nominal parameter value. As the parameter is increased further, it appears that the LM does begin to slightly overpredict the normalized roll reduction factor of the NLM.

From the LM analysis in Eq. (12), it is evident that L and mee have similar sensitivities. Both appear in the same term in the denominator and have a power of one. For this example with the nominal case of L = 0.5 m and mee = 0.5 kg, the lines for the LM sensitivities are very close. However, one can see that if the end effector mass were an order of magnitude larger than the nominal case and the link lengths were smaller or the same as the nominal case discussed, then mee would have a higher sensitivity than L.

This sensitivity analysis also shows that Kϕ has a similar sensitivity to that of L and mee. This is a valuable insight to have because improvements in roll reduction factor similar to that obtained by increasing L and mee can be achieved by only adjusting Kϕ. Adjusting Kϕ changes the control strategy of the manipulator arm, but it does not require changing the arm physical parameters (which may not be feasible in certain situations).

This analysis indicates that the LM gives a good prediction of the expected improvement in reducing roll and load transfer. The LM could be a very useful tool to robot designers in assessing how much of an improvement they could achieve in roll stability for different design scenarios. For example, the robot designer could use the sensitivity plots in Fig. 7 to select a gain Kϕ for their robot. In selecting Kϕ, the designer would want to consider how much torque the arm joint motors can provide. To get an estimate of the torque required, the designer could calculate the steady-state reaction moment due to the manipulator arm Ma for the extreme case when a tire lifts off and multiply by a safety factor, e.g., τmSF·Ma. After using the LM as a starting point for their design, it would then be necessary to test in a higher fidelity simulation similar to the NLM used here.

Rollover Stability Analysis.

While the LM has been limited to only predicting steady-state analysis, the NLM can be used to observe the transient behavior of the system. Figure 8 shows the transient response for a test case of Vx = 8 m/s and steplike steering input of δ = 4 deg.

Notice how the radius of the turn is smaller and lateral acceleration is larger for the Arm Mov. case as shown in Figs. 8(a) and 8(b), respectively. This indicates that moving the manipulator arm during maneuvers does affect the handling dynamics. This trend was not captured by the LM since the handling and roll dynamics were decoupled. This helps explain deviation between the LM and NLM in Fig. 5.

The cause for this change in handling dynamics could be due to the difference in normal load distribution. For Arm Mov., R will decrease indicating a more even distribution of load between the left and right side of the vehicle. Under this condition, the tires are able to generate lateral force more effectively and the vehicle performs a tighter radius turn for a given steering input.

Additionally, the roll angle of the tire influences the tire behavior [28]. As the tire experiences larger roll angles, its ability to generate lateral force decreases. This agrees with the trend shown in Fig. 5 that since the mobile manipulator with Arm Mov. experiences a smaller roll angle ϕ, it is able to more effectively generate lateral acceleration ay and perform a tighter radius turn with the same forward speed and steering input as the mobile manipulator with Arm Stat.

Further exploration of this effect of Arm Mov. on handling dynamics will be discussed in Sec. 5.3. For now, we will consider a set of steering inputs that result in a turn with the same radius as shown in Fig. 9. From Figs. 9(a) and 9(b), it can be seen that the X–Y paths for Arm Mov. and Arm Stat. mobile manipulators are very close, as are the lateral accelerations. Both mobile manipulators had forward speeds of Vx = 8 m/s; however, the mobile manipulator with Arm Mov. only needed a steering input of δ=3deg, while a steering input of δ=6deg was needed with Arm Stat.

From Figs. 8 and 9, we can see the rollover stability improvement from Arm Mov. will appear to be larger when analyzed in terms of forward speed Vx and path radius, rather than in terms of forward speed Vx and steering angle δ. In the analysis for Fig. 8, where both systems are compared in terms of the same forward speed (Vx = 8 m/s) and steering input (δ=4deg), the roll reduction factor is ρδ=0.12. In the analysis for Fig. 9, where both systems are compared in terms of the same forward speed (Vx = 8 m/s) and X–Y path radius (δmoving=3deg,δstat.=6deg), the roll reduction factor is ρradius=0.29. A larger roll reduction is found when comparing the Arm Mov. versus Arm Stat. cases for mobile manipulators traveling similar trajectories instead of receiving the same inputs.

Handling Dynamics Analysis.

In order to gain some insight into how weight-shifting affects the overall stability of the system, a batch of simulations was run with the NLM. A range of forward velocities of Vx = 1–15 m/s was run with steering inputs of δ = 1–15 deg for both cases of Arm Stat. and Arm Mov.

For each maneuver, any combination of tire lift-off (TLO) and tire saturation (TS) can occur. TLO is defined to be when any one of the tire normal forces reaches zero. TS is defined to be when the magnitude of the combined lateral and longitudinal forces generated by the tire reach the friction limit. During simulations of the NLM, the following three conditions occurred:

  1. 1.

    NTLO, NTS: No tire lift-off and no tire saturation

  2. 2.

    NTLO, TS: No tire lift-off and tire saturation

  3. 3.

    TLO, NTS: Tire lift-off and no tire saturation

Data from the simulation runs were checked to see which one of three conditions it exhibited during the transient response. If TLO or TS was detected at any point during the transient, then that combination of forward velocity and steering input was labeled appropriately. Note: once the tires began to saturate, the vehicle was not able generate enough lateral force to cause a TLO event for the turning maneuver tested. Thus, a TLO and TS condition did not occur during any of the simulation runs.

Figure 10 shows the results of this batch of simulations for Arm Stat. (left) and Arm Mov. (right). These graphs show that by moving the manipulator arm as previously described, the stability region of allowable forward speeds and steering inputs increases. Note that in the Arm Mov. Case, the TLO region becomes smaller, however, the NTLO-TS region becomes larger. When the tires saturate, the operator is limited in their ability to generate additional lateral force and control the vehicle. While NTLO-TS is undesirable, it can be easier to recover from than the TLO scenario.

As Fig. 9 shows, a smaller steering input is required to achieve the same radius turn/lateral acceleration for the Arm Mov. versus Arm Stat. case. Thus, viewing the rollover stability region in terms of lateral acceleration versus forward speed is also informative to understand how weight-shifting can improve maneuverability. Figures 10(c) and 10(d) depict the same rollover stability region in Figs. 10(a) and 10(b) in terms of lateral acceleration versus forward speed.

The dashed and solid lines in Fig. 10 are present to aid in visualizing how the rollover stability regions in plots Figs. 10(a) and 10(b) map to the regions in plots Figs. 10(c) and 10(d), respectively. That is, the simulations used to define the rollover stability regions above the dashed line and below the solid line in Figs. 10(a) and 10(b) produced the lateral accelerations above the dashed and below the solid line in Figs. 10(c) and 10(d). Note that for low speeds (less than 4 m/s), steering angles of 15 deg did not result in TLO or TS. Thus, the vehicle can likely still achieve higher lateral accelerations at these low forward speeds by using larger steering angles. Additionally, the smaller lateral accelerations below the dashed lines at high forward speeds in Figs. 10(c) and 10(d) are presumably achievable with steering angles smaller than 1 deg. Overall, it is evident from Fig. 10 that the region of achievable lateral accelerations that result in NTLO is larger for the Arm Mov. case (Figs. 10(a) and 10(c)) in comparison to the Arm Stat. case (Figs. 10(b) and 10(d)).

The impact of weight-shifting on the handling dynamics can also be seen in Fig. 11. For each of the forward speed-steering angle input combinations that resulted in NTLO in Fig. 10, the percent increase in lateral acceleration for the Arm Mov. versus Arm Stat. case is shown in Fig. 11. At low speeds and low steering angles, weight-shifting has a smaller effect on handling. That is, for a constant steering input at constant speed, the resulting lateral acceleration will only be slightly higher for the Arm Mov. case. However, at higher speeds and steering angles, the steering will become more sensitive for the Arm Mov. case. Overall, Fig. 10 illustrates that weight-shifting increases the maximum lateral acceleration we can achieve (allowing for smaller radius turns at high speeds), while Fig. 11 shows that this causes the sensitivity of the steering angle to increase.

Experimental Validation.

To support our analysis with the LM and NLM, we implemented our dynamic weight-shifting control method on the hardware shown in Fig. 1.

Experimental Setup.

The vehicle platform used was a 1:10 scale RC car chassis from Team Associated. A custom two-link manipulator arm was fabricated primarily by a senior design team at the University of Michigan2. The manipulator arm joints were actuated by a Dynamixel MX-64 servo (base-link 1 joint) and a Dynamixel MX-28 servo (link 1-link 2 Joint). The Dynamixel servos have built in PID controllers allowing for easy position control.

Onboard the experimental platform was a Raspberry Pi model B microcomputer. The microcomputer collected sensor data from an inertial measurement unit (IMU) and commanded the manipulator arm joints to the desired angular positions. We used a 6DOF IMU, consisting of a three-axis accelerometer (ADXL345) and three-axis rate gyro (ITG3200), to estimate vehicle states. Data were collected from the IMU at 100 Hz. Roll angle was estimated in real time on the microcomputer using a moving average filter consisting of ten points. Based on the estimated roll angle, joint angle position commands were sent to the arm at 10 Hz.

The dimensions and weight of the experimental platform were very close to those used for the LM and NLM. One difference was that the manipulator links on the experimental platform were slightly shorter, link 1 was approximately 0.3 m, and link 2 was 0.2 m.

The experimental platform was operated outdoors in a flat, open parking lot. A camera was placed at a high vantage point overlooking the parking lot and captured the movement of the vehicle at 30 Hz. The camera was calibrated and the homography transformation matrix relating the pixel data to world frame data was calculated. A red marker was placed on the back of the vehicle so that it could be easily tracked using simple color threshold detection in the video recorded for each maneuver. This allowed us to estimate the x–y position and speed of the vehicle during each maneuver.

Experimental Results.

Two comparisons will be made between our experimental results and analysis with the models. The first supports the effect of the arm on preventing rollover and the second supports analysis of the arm's effect on handling dynamics.

The same type of maneuver performed with the NLM in Figs. 8 and 9 was performed with the experimental platform. Given the size of the parking lot we tested in and the capabilities of the hardware, we were able to perform the steplike steering input maneuver at a speed of just over 5 m/s. The magnitude of the steering input was 20 deg, which was larger than the magnitude of the steering inputs applied to the NLM. We chose a larger magnitude steering input to keep the radius of the maneuver small while still experiencing large lateral accelerations that occur in a rollover event. The manipulator arm joint angle constant was set to Kϕ=2 for the Arm Mov. case.

The results of the turning maneuver with the experimental platform are shown in Fig. 12. The x–y position data collected using computer vision is shown in subplot Fig. 12(a). Markers are included at each 0.5 s interval to compare with the time series data in subplots Figs. 12(b)12(d). All IMU signals were filtered using a fourth-order Butterworth filter with cutoff frequency 5 Hz and matlab's zero-phase distortion filtering function filtfilt. The roll angle in subplot Fig. 12(c) was estimated using x-IO Technologies' implementation3 of Mahony's Attitude and Heading Reference System (AHRS) algorithm [29]. During the maneuver recorded in Fig. 12, the vehicle's wheels actually lifted off the ground and the vehicle appeared to roll just past the point at which the outriggers first hit the ground for the Arm Stat. case. We measured that the outriggers (see Fig. 1) touch the ground when the vehicle is rolled approximately 42 deg in the static case. Thus, the AHRS algorithm's gain was tuned such that the maximum roll angle of the Arm Stat. case was just under 50 deg. The lateral and vertical accelerations were then transformed from the vehicle frame to the world frame via multiplication by a rotation matrix about the vehicle's roll axis.

From Fig. 12(c), we can see that moving the manipulator arm with weight-shifting had a dramatic effect on reducing the vehicle's roll angle. In this case, when the vehicle performed the maneuver with the Arm Stat., the tires lifted off and the outriggers hit the ground. This event occurred for around 1.5 s. We can see in subplot Fig. 12(c) that this is where the roll angle peaks and in subplot Fig. 12(d) the vertical acceleration goes positive (in the static case, gravity is −1 g in the z-direction).

The second comparison made with the experimental results supports the handling dynamics analysis. The vehicle was given a constant steering input of 15 deg and a constant throttle input of 45% max throttle, until it reached steady-state driving in a circle. The resulting maneuver with the experimental platform is shown in Fig. 13.

The vehicle's forward speed was estimated using video data. In particular, the radius of the circle driven by the vehicle was estimated and the time the vehicle took to complete a lap was calculated. In steady-state, the forward speed of the vehicle for the Arm Stat. case was approximately 3.8 m/s and the resulting average lateral acceleration for time 2–8 s was 0.802 g. The forward speed for the Arm Mov. case was 4.3 m/s with an average lateral acceleration of 0.906 g. Comparing these two lateral accelerations gives a 12.9% increase in lateral acceleration with the Arm Mov. versus Arm Stat. case. This is similar to the value of 17.6% found with the NLM at forward speed 4 m/s and steering angle 15 deg (Fig. 11). Overall, the experimental results appear to line up well with results from the NLM simulations. Both effects of reducing roll angle and increasing lateral acceleration gain were reproduced on the experimental platform.

Conclusions

This paper presented a control method that uses an existing manipulator arm to improve rollover stability and increase maneuverability in high-speed maneuvers. The LM was shown to be a useful tool for analyzing the sensitivity of design parameters including link length L, mass mee, and joint angle constant Kϕ. The NLM was used to describe the impact of dynamic weight-shifting on both roll dynamics and maneuverability. Experimental results from a hardware implementation of dynamic weight-shifting supported both relationships for roll dynamics and maneuverability developed with the NLM. Unlike other dynamic stability control methods that require adding additional hardware, decreasing vehicle speed or increasing turn radius, the results of this paper demonstrated that by dynamically shifting the weight of the manipulator arm, turns can be taken at a higher-speed and smaller radius compared to keeping the manipulator arm static.

Dynamic weight-shifting can safely increase speed and maneuverability when operating mobile manipulators due to the improved rollover stability. Results from our analysis suggest several areas of future work. One could investigate a linearized model that captures features like the interaction between roll and handling dynamics, or tripped rollover events. These linearized models could be used in model-based control methods, such as model predictive control, to further improve UGV performance. One could also consider how to use the stabilizing moments due to gravity (described in this paper) in conjunction with stabilizing moments due to reaction torques of the manipulator arm's motion (described in prior works). That is, the manipulator arm control method may switch between the two strategies based on the type of maneuver or impending rollover (untripped versus tripped).

Acknowledgment

The authors thank Dr. Jimmy Chiu for contributing files of the nonlinear SimMechanics vehicle model [16] and the senior design team for their help fabricating the experimental platform. Thank you to Steve Vozar for his helpful insights. This research was supported in part by a fellowship from the Department of Mechanical Engineering and the Automotive Research Center at the University of Michigan, with funding from Government Contract No. DoD-DoA W56HZV-14-2-0001 through the U.S. Army Tank Automotive Research, Development, and Engineering Center.

Appendix: Model Parameter Description

A brief description of parameters and values used in the LM and NLM are listed in Table 1. There are differences in tire stiffness kt and damping bt values for the LM and NLM because the NLM contains roll stiffness and damping both in the tires and about its roll center axis. The LM contains all of its roll stiffness and damping in the tires. The NLM tires are stiff compared to the roll stiffness. Therefore, kroll and broll decrease the overall effective stiffness and damping. Smaller values of kt and bt were selected for the LM in order to match the steady-state response of the NLM in Fig. 5 for u = 4 m/s.

The NLM inertia parameters of the body, front axle, rear axle, front wheel, and rear wheel were the following (with units (10−4 kg/m2)): 
Ibody=diag([227.5, 1340, 1609])Iax,f=diag([5.647, 0.1613, 5.638])Iax,r=diag([5.748, 0.1642, 5.739])Iwh,f=diag([0.1777, 0.3080, 0.1777])Iwh,r=diag([0.1470, 0.2547, 0.1470])
(A1)

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