Tolerance　design　of　key　components　is　an　effective　method　to　improve　the　machine　tools　accuracy.　Currently,　tolerance　design　highly　depends　on　the　personal　experience,　which　often　subjectively　determines　the　machine　tool　accuracy　and　also　increases　the　manufacturing　costs.　To　avoid　the　phenomenon,　a　machining　error　prediction　model　(MEPM)　assisting　tolerance　design　is　proposed　and　experimental　verified　based　on　a　precision　vertical　grinding　machine　(PVGM).　Firstly,　the　mapping　relationship　between　the　tolerance　parameter　and　geometric　error　of　key　components　is　obtained　as　the　superimposed　Fourier　series.　Subsequently,　the　volumetric　error　model　including　geometric　errors　is　established　by　the　homogeneous　transform　matrix　and　multi-body　system　theory.　Based　on　a　typical　workpiece,　a　tolerance-based　MEPM　is　obtained　by　integrating　the　ideal　machining　path.　Finally,　a　grinding　test　on　PVGM　is　conducted　to　validate　the　MEPM.　The　results　show　that　the　predicted　roundness　of　three　positions　is　3.033 μm,　2.905 μm,　and　2.774 μm,　respectively,　and　the　measured　roundness　are　3.163 μm,　2.980 μm,　and　2.904 μm,　respectively.　The　error　between　PVGM　predicted　and　measured　roundness　is　not　more　than　5%.　The　MEPM　is　important　for　assisting　the　tolerance　design　and　improving　the　machining　accuracy　in　the　machine　tools.　©　The　Author(s),　under　exclusive　licence　to　Springer-Verlag　London　Ltd.,　part　of　Springer　Nature　2024.