METHOD FOR DETERMINING A MASS OF AN ATTACHED IMPLEMENT FOR A UTILITY VEHICLE

Abstract

A method for determining a mass of an implement includes providing a rear powerlift having at least one upper link and at least one lower link and a support structure disposed at the rear of a utility vehicle. The method includes defining an angle (Ψ) between the upper link and a vehicle horizontal line, an angle (φ) between the lower link and the vehicle horizontal line, an angle of inclination (θ) of a vehicle horizontal line relative to a terrestrial horizontal line, a path (LV) representative of a connection along the lower link between the support structure and the implement, and a force (F.sub.U) impinging on a connection between the upper link and the implement and acting along the upper link. The mass is determined by at least one of the angle (Ψ), the angle (φ), the angle of inclination (θ), the path (LV), and the force (F.sub.U).

Claims

1. A method for determining a mass of an implement of a utility vehicle, comprising: providing a rear powerlift having at least one upper link and at least one lower link, a support structure disposed at the rear of the utility vehicle, and the implement being articulatably coupled to the support structure; defining an angle (Ψ) between the upper link and a vehicle horizontal line, an angle (φ) between the lower link and the vehicle horizontal line, an angle of inclination (θ) of a vehicle horizontal line relative to a terrestrial horizontal line, a path (LV) representative of a connection along the lower link between the support structure and the implement, and a force (F.sub.U) impinging on a connection between the upper link and the implement and acting along the upper link; and determining the mass of the implement as a function of at least one of the angle (Ψ), the angle (φ), the angle of inclination (θ), the path (LV), and the force (F.sub.U).

2. The method of claim 1, further comprising: providing the rear powerlift with an adjustable-length lifting arm; and determining the mass of the attached implement as a function of an angle (ρ) that is enclosed by a vehicle-vertical line and a connecting path (RT) between two operative ends of the lifting arm.

3. The method of claim 2, further comprising determining the mass of the attached implement as a function of a force (F.sub.T) acting between the two operative ends of the lifting arm.

4. The method of claim 2, wherein a first operative end of the lifting arm comprises an articulated connection to the support structure, and a second operative end of the lifting arm comprises an articulated connection to a link-connecting point of the lower link.

5. The method of claim 4, further comprising determining the mass of the attached implement as a function of a path (LT) defined by a connection along the lower link between the support structure and the link-connecting point.

6. The method of claim 4, wherein the first operative end comprises an articulated connection via a joint arm to the support structure.

7. The method of claim 6, wherein the joint arm comprises: a first articulation point associated with the support structure and a second articulation point associated with the lifting arm; and a third articulation point between the first and second articulation points.

8. The method of claim 7, further comprising providing another adjustable-length lifting arm comprising an articulated connection to the third articulation point and the support structure.

9. The method of claim 2, further comprising providing a lifting arm including a piston-cylinder unit.

10. The method of claim 2, wherein the lift arm comprises a spindle-thread unit.

11. The method of claim 1, further comprising determining a center of gravity of the attached implement as a function of the determined mass of the attached implement.

12. The method of claim 1, further comprising: defining a coordinate system having an x-axis oriented parallel to a longitudinal direction of the utility vehicle and a z-axis oriented parallel to a vertical direction of the utility vehicle, wherein the x-axis and the z-axis intersect in a zero point of the coordinate system; and determining the mass or a center of gravity of the attached implement based on the coordinate system.

13. The method of claim 12, further comprising arranging a zero point on a rear axle of the utility vehicle.

14. The method of claim 8, further comprising determining an x-coordinate of the center of gravity of the attached implement as a function of the mass of the attached implement.

15. The method of claim 14, wherein the determining the x-coordinate of the center of gravity step is determined as a function of at least one of the angle (Ψ) between the upper link and a vehicle horizontal line, the force (F.sub.U) impinging on a connection between the upper link and the attached implement and acting along the upper link, an x-coordinate of an articulation point of the lower link on the attached implement, a difference between the x-coordinates of an articulation point of the upper link on the attached implement and an articulation point of the lower link on the attached implement, and a difference between the z-coordinates of an articulation point of the upper link on the attached implement and an articulation point of the lower link on the attached implement.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0037] The above-mentioned aspects of the present disclosure and the manner of obtaining them will become more apparent and the disclosure itself will be better understood by reference to the following description of the embodiments of the disclosure, taken in conjunction with the accompanying drawings, wherein:

[0038] FIG. 1 is a schematic side view of an attached implement articulated to a rear powerlift at the front end of a utility vehicle;

[0039] FIG. 2 is a schematic representation of forces impinging on a joint arm of the rear powerlift according to FIG. 1;

[0040] FIG. 3 is a schematic representation of forces impinging on the attached implement from FIG. 1; and

[0041] FIG. 4 is a schematic representation of forces impinging on a lower link of the rear powerlift according to FIG. 1.

DETAILED DESCRIPTION

[0042] In FIG. 1, an attached implement 10 is schematically shown articulated to the rear end of a utility vehicle (such as a tractor). A powerlift 12, designed as a rear powerlift 12, is articulated onto a support structure 14 of the utility vehicle. The rear powerlift 12 has an upper link 16 and two parallel lower links 18 for receiving the attached implement 10. The upper link 16 has an articulated connection via an articulation point W to the support structure 14 and via an articulation point U to the attached implement 10. In a transverse direction of the utility vehicle, running perpendicular to the plane of FIG. 1, the upper link 16 is arranged centrally between two parallel lower links 18. Each lower link 18 has an articulated connection via an articulation point L to the support structure 14 and via an articulation point V to the attached implement 10. The articulation points U, V on the attached implement 10 are designed in the usual manner, e.g., as catch hooks for the rear powerlift 12.

[0043] In the present embodiment, the articulation point W permits three different positions in the vertical direction 20 for articulating the upper link 16. The respective position is defined by a user and appropriately installed. An angle Ψ is enclosed between the upper link 16 and a vehicle horizontal line 22. An angle φ is enclosed between each lower link 18 and a vehicle horizontal line 22.

[0044] The angles formed between the vehicle horizontal line 22 and a vehicle vertical line 32, particularly angles Ψ and φ, relate to a fixed vehicle coordinate system 33 having an x-axis and a z-axis. The x-axis runs parallel to a longitudinal direction 24 of the utility vehicle or parallel to the vehicle horizontal line 22. The z-axis runs parallel to the vertical direction 20 of the utility vehicle or parallel to the vehicle vertical line 32. The inclination of the support structure 14, and thus of the vehicle horizontal line 22 of the utility vehicle, relative to the terrestrial horizontal line 21, is represented by an angle of inclination θ. The angle of inclination θ takes on values greater than zero when the utility vehicle is oriented uphill in the forward direction. The angle of inclination θ analogously assumes values less than zero when the utility vehicle is oriented downhill in the forward direction.

[0045] A lifting arm designed in a conventional manner as an adjustable-length lifting spindle 36 (i.e., a spindle-thread unit) is a component of the rear powerlift 12. The lifting spindle 36 has two operative ends 38, 40. One operative end 38 has an articulated connection to an articulation point R of a joint arm 42, whereas the other operative end 40 of the lifting spindle 38 is articulated to a link-connecting point T of the lower link 18. In the present embodiment, the link-connecting point T allows three different positions along the longitudinal direction 24 for articulating the lifting spindle 36 to the lower link 18. The respective position is defined by a user and installed accordingly. A connecting path between the two operative ends 38, 40 of the lifting spindle 36 encloses an angle ρ with the vehicle vertical line 32 running in the vertical direction 20. Both lower links 18 are connected to a lifting spindle 36 in the manner described.

[0046] The joint arm 42 has a first articulation point P for an articulated connection to the support structure 14. At the second articulation point R, the joint arm 42 has an articulated connection to the lifting spindle 36. A third articulation point N, which is arranged on an arm 44 of the joint arm 42, is present between the two articulation points P, R. The arm 44 branches off at a branching point Z of the path PR. An additional lifting arm in the form of a lifting cylinder 46 has an articulated connection to the third articulation point N and the support structure 14 at the articulation point M. A lifting spindle 36, a joint arm 42 and a lifting cylinder 46 are provided for each lower link 18.

[0047] The connecting path between the articulation points P and R encloses an angle δ with the vehicle horizontal line 22. The connecting path between the articulation points P and N encloses an angle ζ with the vehicle horizontal line 22. The connecting path between the two operative ends of the lifting spindle 46 encloses an angle α with the vehicle vertical line 32.

[0048] For mathematical/geometrical determination of a mass m and a center of gravity S of the rear attached implement 10, a coordinate system 33 is defined, the x-axis of which is oriented parallel to the longitudinal direction 24 and the z-axis of which is oriented parallel to the vertical direction 20. In the present embodiment of FIG. 1, the zero point of this coordinate system 33 lies on a schematically indicated rear axle 48 of the utility vehicle. A rear wheel 50 of the rear axle 48 is also schematically shown.

[0049] FIGS. 2-4 present different forces impinging on the force system between support structure 14 and attached implement 10. A force of which the x-component is designated F.sub.Px and the z-component is designated F.sub.Pz impinges on the articulation point P (FIG. 2). A cylinder force F.sub.Zy1 is active between the two operative ends of the lifting cylinder 46. A force F.sub.T is active between the articulation point R of the joint arm 42 and the link-connecting point T of the lower link 18. A force F.sub.V, which is oriented at an angle β relative to a vehicle vertical line 32, impinges on the articulation point V (FIG. 3). A force impinging on the articulation point U and acting along the upper link 16 is designated F.sub.U. The weight force of the attached implement 10 at center of gravity S is marked m.Math.g, where g is the gravitational constant. A force of which the x-component is designated F.sub.Lx and the z-component is designated F.sub.Lz impinges on the articulation point L (FIG. 4).

[0050] Based on the torque and force equilibria in FIG. 2, the following relationships can be assumed at the rear powerlift 12:

ΣM.sub.i.sup.(P)=0.fwdarw.0=−PR.Math.{right arrow over (F)}.sub.T.Math.cos(δ+ρ)+PN.Math.{right arrow over (F)}.sub.Zy1.Math.cos(ζ−α)  (1)

Σ{right arrow over (F)}.sub.ix=0.fwdarw.0={right arrow over (F)}.sub.Px+{right arrow over (F)}.sub.Zy1.Math.sinα {right arrow over (F)}.sub.T.Math.sin ρ  (2)

Σ{right arrow over (F)}.sub.iz=0.fwdarw.0={right arrow over (F)}.sub.Pz+{right arrow over (F)}.sub.Zy1.Math.cosα−{right arrow over (F)}.sub.T.Math.cos ρ  (3)

[0051] It also follows from equation (1) that

[00001]$\begin{array}{cc}{\stackrel{.fwdarw.}{F}}_T={\stackrel{.fwdarw.}{F}}_{\mathrm{Zyl}}.Math.\frac{\stackrel{\_}{\mathrm{PN}}}{\stackrel{\_}{\mathrm{PR}}}.Math.\frac{\cos .Math..Math.\left(\zeta -\alpha \right)}{\cos .Math..Math.\left(\delta +\rho \right)}& \left(4\right)\end{array}$

[0052] Due to the torque and force equilibria in FIG. 3, the following additional relationships at the rear powerlift 12 can be assumed:

ΣM.sub.i.sup.(V)=0.fwdarw.0=(U.sub.x−V.sub.x).Math.{right arrow over (F)}.sub.U.Math.sinΨ+(U.sub.z−V.sub.z).Math.{right arrow over (F)}.sub.U.Math.cosΨ−(S.sub.x−V.sub.x).Math.m.Math.g.Math.cosθ−(S.sub.z−V.sub.z).Math.m.Math.g.Math.sinθ  (5)

Σ{right arrow over (F)}.sub.ix=0.fwdarw.0={right arrow over (F)}.sub.V.Math.sin β−{right arrow over (F)}.sub.U.Math.cosΨ+m.Math.g.Math.sinθ.fwdarw.{right arrow over (F)}.sub.V.Math.sin β={right arrow over (F)}.sub.U.Math.cosΨ−m.Math.g.Math.sinθ  (6)

Σ{right arrow over (F)}.sub.iz=0.fwdarw.0={right arrow over (F)}.sub.V.Math.cos β−{right arrow over (F)}.sub.U.Math.sinΨ+m.Math.g.Math.cosθ.fwdarw.{right arrow over (F)}.sub.V.Math.cos β={right arrow over (F)}.sub.U.Math.sinΨ−m.Math.g.Math.cosθ  (7)

[0053] Due to the torque and force equilibria in FIG. 4, the following additional relationships at the rear powerlift can be assumed:

ΣM.sub.i.sup.(O)=0.fwdarw.0=−LT.Math.{right arrow over (F)}.sub.T.Math.sin ρ.Math.sinφ+LT.Math.{right arrow over (F)}.sub.T.Math.cos ρ.Math.cosφ−LV.Math.{right arrow over (F)}.sub.V.Math.sin φ.Math.sin β−LV.Math.{right arrow over (F)}.sub.V.Math.cos φ.Math.cos β  (8)

Σ{right arrow over (F)}.sub.iz=0.fwdarw.0={right arrow over (F)}.sub.V.Math.sin β−{right arrow over (F)}.sub.T.Math.sin ρ+{right arrow over (F)}.sub.Lz  (9)

Σ{right arrow over (F)}.sub.iz=0.fwdarw.0={right arrow over (F)}.sub.V.Math.cos β−{right arrow over (F)}.sub.T.Math.cos ρ+{right arrow over (F)}.sub.Lz  (10)

[0054] By inserting equations (6) and (7) into equation (8), and by solving equation (8) for the mass m of the attached implement 10, it follows that for the mass m

[00002]$\begin{array}{cc}m=\frac{\stackrel{\_}{\mathrm{LT}}.Math.{\stackrel{.fwdarw.}{F}}_T.Math.\cos .Math..Math.\left(\varphi +\rho \right)-\mathrm{LV}.Math.{\stackrel{.fwdarw.}{F}}_U.Math.\sin .Math..Math.\left(\varphi -\psi \right)}{\stackrel{\_}{\mathrm{LV}}.Math.g.Math.\cos .Math..Math.\left(\varphi +\theta \right)}& \left(11\right)\end{array}$

Thus, the mass m is determined as a function of [0055] the angle Ψ between the upper link 16 and the vehicle horizontal line 22, [0056] the angle φ between the lower link 18 and the vehicle horizontal line 22, [0057] the angle ρ between the lifting spindle 36 and the vehicle vertical line 32, [0058] the angle of inclination θ of the support structure 14 or the vehicle horizontal line 22 of the utility vehicle in relation to the terrestrial horizontal line 21, [0059] the path LV along the lower link 18 as the connection between the articulation points L and V, [0060] the path LT along the lower link 18 between the articulation point L and the link-connecting point T, [0061] the force F.sub.T on the lifting spindle 36, and [0062] the force F.sub.U along the upper link 16.

[0063] The length of the path LV is a known design parameter of the rear powerlift 12. The length of the path LT can be measured by means of a length sensor or a distance sensor, for example, or the length of this path LT is known based on the user-dependent installation of the lifting spindle 36 on the lower link 18. The force F.sub.U can be measured by means of pressure sensors on the upper link 16 or can alternatively by measured means of a biaxial force sensor at the articulation point W. The force F.sub.T can be indirectly determined by measuring the force F.sub.Zy1 at the lifting cylinder 46 and can be taken into account in equation (4). The paths PN and PR of the joint arm 42 in equation (4) are constant and to that extent are known design parameters.

[0064] The angle δ in equation (4) can be measured by means of a suitable sensor (e.g. an angle sensor). From this, the value of angle ζ can be easily derived:

[00003]$\zeta ={\tan }^{-1}\left(\frac{P_z-N_z}{P_x-N_x}\right)$

[0065] The articulation point P has fixed x-and z-coordinates P.sub.x and P.sub.z relative to the coordinate system 33. The x-coordinate N.sub.x and the z-coordinate N.sub.x of the articulation point N follow from

N.sub.x=P.sub.x+PZ.Math.cosδ+ZN.Math.sinδ and N.sub.z=P.sub.z−PZ.Math.sinδ−ZN.Math.cosδ,

where the angle δ, as already mentioned, is measured and the paths PZ, ZN are constant design parameters of the joint arm 42.

[0066] The angle α in equation (4) is determined as follows:

[00004]$\alpha ={\tan }^{-1}\left(\frac{N_x-M_x}{N_z-M_z}\right)$

[0067] The x-coordinate N.sub.x and the z-coordinate N.sub.z of the articulation point N are derived as just explained, while the articulation point M has fixed x- and z-coordinates M.sub.x and M.sub.z relative to the coordinate system 33.

[0068] The angles ρ, φ and Ψ in equations (4) and (11) can be derived as follows:

[00005]$\varphi ={\tan }^{-1}\left(\frac{L_z-T_z}{L_x-T_x}\right)$$\psi ={\tan }^{-1}\left(\frac{U_z-W_z}{U_x-W_x}\right)={\tan }^{-1}\left(\frac{{\stackrel{.fwdarw.}{F}}_{\mathrm{Wz}}}{{\stackrel{.fwdarw.}{F}}_{\mathrm{Wx}}}\right)$$\rho ={\tan }^{-1}\left(\frac{T_x-R_x}{T_z-R_z}\right)$

[0069] The x-coordinate L.sub.x and the z-coordinate L.sub.z are known since the articulation point L thereof on the support structure 14 has fixed coordinates relative to the coordinate system 33. The x-coordinate W.sub.x and the z-coordinate W.sub.z of the articulation point W are also known, depending on the application case, and are therefore either determined by sensors or specified by the operator.

[0070] The above-mentioned angles can also be measured by means of suitable angle sensors. Alternatively, biaxial force measuring pins can be used, as illustrated with reference to angle Ψ and the two force components F.sub.Wx (along the x-axis of the coordinate system 33) and F.sub.Wz (along the z-axis of the coordinate system 33) in schematic form (FIG. 2).

[0071] The variable x-coordinate R.sub.x and z-coordinate R.sub.z of the articulation point R of the joint arm 42 are preferably derived as follows:

R.sub.x=P.sub.x+PR.Math.cosδ and R.sub.z=P.sub.z−PR.Math.sinδ

[0072] The articulation point P has fixed coordinates P.sub.x and P.sub.z relative to the coordinate system 33. The path PR is a fixed design parameter of the joint arm 42.

[0073] The variable x-coordinates and z-coordinates T.sub.x and T.sub.z of the link-connecting point T and U.sub.x and U.sub.z of the articulation point U can be derived mathematically as follows:

[0074] To calculate the link-connecting point T, two circles are defined. The first circle has a radius corresponding to the path LT with a circle center L and the second circle has a radius corresponding to the path RT with a circle center R. The associated circle equations are

(T.sub.x−L.sub.x).sup.2+(T.sub.z−L.sub.z).sup.2=LT.sup.2 (T.sub.x−R.sub.x).sup.2+(T.sub.z−R.sub.z).sup.2=TR.sup.2

[0075] The two circle equations are solved for the z-component of the link-connecting point T. This yields a straight-line equation of a straight line running through the two circle centers, of the form

[00006]$T_z=-\underset{\underset{n}{}}{\left(\frac{R_x-L_x}{R_z-L_z}\right)}.Math.T_x+\underset{\underset{b}{}}{\left(\frac{{\stackrel{\_}{\mathrm{CT}}}^{.Math.2}-{\stackrel{\_}{\mathrm{TR}}}^{.Math.2}+R_x^2+R_z^2-L_x^2-L_z^2}{2.Math.\left(R_z-L_z\right)}\right)}$$T_x=-n.Math.T_x+b$

[0076] With respect to the link-connecting point T, the quadratic equation that follows by inserting the straight-line equation into one of the two circle equations

[00007]$T_x^2+\underset{\underset{p_1}{}}{\frac{2.Math.\left(-R_x-n.Math.w\right)}{1+n^2}}.Math.T_x+\underset{\underset{q_1}{}}{\frac{R_x^2+w^2-{\stackrel{\_}{\mathrm{TR}}}^{.Math.2}}{1+n^2}}=0$

and the auxiliary variables n, b and w

[00008]$n=\left(\frac{R_x-L_x}{R_z-L_z}\right)$$b=\left(\frac{{\stackrel{\_}{\mathrm{LT}}}^{.Math.2}-{\stackrel{\_}{\mathrm{TR}}}^{.Math.2}+R_x^2+R_z^2-L_x^2-L_z^2}{2.Math.\left(R_z-L_z\right)}\right)$$w=b-R_z$

are defined. This yields the following equations for the x-coordinate T.sub.x and the z-coordinate T.sub.z:

[00009]$\begin{array}{cc}{T}_x=-\frac{p_1}{2}+\sqrt{\frac{p_1^2}{4}-q_1}=\frac{\left(R_x+n.Math.w\right)}{1+n^2}+\sqrt{{\left(\frac{\left(R_x+n.Math.w\right)}{1+n^2}\right)}^2-\frac{R_x^2+w^2-{\stackrel{\_}{\mathrm{TR}}}^{.Math.2}}{1+n^2}}& \left(12\right)\\ .Math.T_z=-n.Math.T_x+b& \left(13\right)\end{array}$

[0077] The path TR can be measured by means of a length sensor, for example, or a distance sensor on the lifting spindle 36. Alternatively, the length of this path TR is known based on a specification by the user for the respective setting of the rear powerlift 12.

[0078] Two additional circles are defined for calculating the articulation point U. The first circle has a radius corresponding to the path UV with circle center V and the second circle has a radius corresponding to the path WU with circle center W. The associated circle equations are

(U.sub.x−V.sub.x).sup.2+(U.sub.z−V.sub.z).sup.2=UV.sup.2 (U.sub.x−W.sub.x).sup.2+(U.sub.z−W.sub.z).sup.2=WU.sup.2

[0079] The two circle equations are solved for the z component of the articulation point U. This yields a straight-line equation of a straight line running through the two circle centers of the form

[00010]$U_z=-u.Math.U_x+s$$U_z=-\underset{\underset{u}{}}{\left(\frac{W_x-V_x}{W_z-V_z}\right)}.Math.U_x+\underset{\underset{s}{}}{\left(\frac{{\stackrel{\_}{\mathrm{UI}}}^{.Math.2}-{\stackrel{\_}{\mathrm{WU}}}^{.Math.2}+W_x^2+W_z^2-V_x^2-V_z^2}{2.Math.\left(W_z-V_z\right)}\right)}$

[0080] With respect to the articulation point U, the quadratic equation that follows by inserting the straight-line equation into one of the two circle equations

[00011]$U_x^2+\underset{\underset{p_2}{}}{\frac{2.Math.\left(-W_x-u.Math.z\right)}{1+u^2}}.Math.U_x+\underset{\underset{q_2}{}}{\frac{W_z^2+z^2-{\stackrel{\_}{\mathrm{WU}}}^{.Math.2}}{1+u^2}}=0$

as well as the auxiliary variables u, s and z

[00012]$u=\left(\frac{W_x-V_x}{W_z-V_z}\right)$$s=\left(\frac{{\stackrel{\_}{\mathrm{UV}}}^{.Math.2}-{\stackrel{\_}{\mathrm{WU}}}^{.Math.2}+W_x^2+W_z^2-V_x^2-V_z^2}{2.Math.\left(W_z-V_z\right)}\right)$$z=s-W_z$

are defined. This yields the following equations for the x-coordinate U.sub.x and the z-coordinate U.sub.z:

[00013]$\begin{array}{cc}{U}_x=-\frac{p_2}{2}+\sqrt{\frac{p_2^2}{4}-q_2}=\frac{\left(W_x+u.Math.z\right)}{1+u^2}+\sqrt{{\left(\frac{\left(W_x+u.Math.z\right)}{1+u^2}\right)}^2-\frac{W_x^2+z^2-{\stackrel{\_}{\mathrm{WU}}}^{.Math.2}}{1+u^2}}& \left(14\right)\\ .Math.U_z=-u.Math.U_x+s& \left(15\right)\end{array}$

[0081] The path WU can be measured by means of a length sensor or a distance sensor on the upper link 16, for example. The path UV is a defined value corresponding to the dimensioning of the attached implement 10.

[0082] The x-coordinate S.sub.x of the center of gravity S of the attached implement 10 relative to the coordinate system 33 can be determined by solving equation (5) for S.sub.x. In this case, the angle of inclination θ is assumed to be 0°:

[00014]$\begin{array}{cc}{S}_x=\frac{1}{m.Math.g}\left[\left(U_x-V_x\right).Math.{\stackrel{.fwdarw.}{F}}_U.Math.\sin .Math..Math.\psi +\left(U_z-V_z\right).Math.{\stackrel{.fwdarw.}{F}}_U.Math.\cos .Math..Math.\psi +V_x.Math.m.Math.g\right]& \left(16\right)\end{array}$

[0083] Thus the x-coordinate of the center of gravity S is determined as a function of [0084] the determined mass m, [0085] the angle Ψ between the upper link 16 and the vehicle horizontal line 22, [0086] the force F.sub.U on the upper link 16, [0087] the x-coordinate V.sub.x of the articulation point V of the lower link 18 at the attached implement 10, [0088] the difference between the x-coordinates U.sub.x and V.sub.x of the articulation points U and V at the attached implement 10, and [0089] the difference between the z-coordinates U.sub.z and V.sub.z of the articulation points U and V at the attached implement 10.

[0090] The x-coordinate V.sub.x and the z-coordinate V.sub.z of the articulation point V follow from, for example

V.sub.x=L.sub.x+LV.Math.cosφ and V.sub.z=L.sub.z−LV.Math.sinφ,

wherein the path LV is a known design parameter and the articulation point L has fixed x- and z-coordinates L.sub.x and L.sub.z relative to the coordinate system 33. The other components and values of equation (16) can be derived according to the above description.

[0091] While embodiments incorporating the principles of the present disclosure have been described hereinabove, the present disclosure is not limited to the described embodiments. Instead, this application is intended to cover any variations, uses, or adaptations of the disclosure using its general principles. Further, this application is intended to cover such departures from the present disclosure as come within known or customary practice in the art to which this disclosure pertains and which fall within the limits of the appended claims.