Solving System of Equations with Vector Relation

Hello,

I am trying to write a program to calculate a given robot's orientation when it is placed on a given terrain in 2D. The robot is wheeled and one of the contact points with the terrain is given as an input. It can essentially be treated as a kinematic chain. The terrain profile is described by two vectors (x and y for each respective dimension).

To solve for the rover configuration in space, I need to solve a system of kinematic equations. However, without taking the relation between each contact point's x and y dimensions (based on the road profile), the system is under constrained. What is the best way to implement the relation of the two vectors into my system of equations? Is there a closed form method of doing this or do I have to implement a solution by iteration? Do I need to perform gradient descent on each contact point to match it with the terrain profile?

Thanks!

Max